Heat capacity is a thermophysical characteristic that determines the ability of bodies to give or receive heat in order to change the body temperature. The ratio of the amount of heat supplied (or removed) to this process, to a change in temperature is called the heat capacity of the body (system of bodies): C \u003d dQ / dT, where is the elementary amount of heat; - an elementary change in temperature.

The heat capacity is numerically equal to the amount of heat that must be supplied to the system in order to increase its temperature by 1 degree under given conditions. The unit of heat capacity is J/K.

Depending on the quantitative unit of the body to which heat is supplied in thermodynamics, mass, volume and molar heat capacities are distinguished.

Mass heat capacity is the heat capacity per unit mass of the working fluid, c \u003d C / m

The unit of mass heat capacity is J/(kg×K). Mass heat capacity is also called specific heat capacity.

Volumetric heat capacity is the heat capacity per unit volume of the working fluid, where and are the volume and density of the body under normal physical conditions. C'=c/V=c p . Volumetric heat capacity is measured in J / (m 3 × K).

Molar heat capacity - heat capacity, related to the amount of the working fluid (gas) in moles, C m = C / n, where n is the amount of gas in moles.

Molar heat capacity is measured in J/(mol×K).

Mass and molar heat capacities are related by the following relation:

The volumetric heat capacity of gases is expressed in terms of molar as

Where m 3 / mol is the molar volume of gas under normal conditions.

Mayer equation: C p - C v \u003d R.

Considering that the heat capacity is not constant, but depends on temperature and other thermal parameters, a distinction is made between true and average heat capacity. In particular, if you want to emphasize the dependence of the heat capacity of the working fluid on temperature, then write it as C(t), and specific - as c(t). Usually, true heat capacity is understood as the ratio of the elementary amount of heat that is reported to a thermodynamic system in any process to an infinitesimal increase in the temperature of this system caused by the imparted heat. We will consider C(t) the true heat capacity of the thermodynamic system at a system temperature equal to t 1 , and c(t) - the true specific heat capacity of the working fluid at its temperature equal to t 2 . Then the average specific heat of the working fluid when its temperature changes from t 1 to t 2 can be determined as

Usually, the tables give the average values ​​of the heat capacity c cf for various temperature intervals, starting from t 1 \u003d 0 0 C. Therefore, in all cases when the thermodynamic process takes place in the temperature range from t 1 to t 2, in which t 1 ≠ 0, the amount The specific heat q of the process is determined using the tabular values ​​of the average heat capacities c cf as follows.

The perfection of thermal processes occurring in the cylinder of a real automobile engine is evaluated by the indicator indicators of its actual cycle, while the perfection of the engine as a whole, taking into account power losses due to friction and the drive of auxiliary mechanisms, is evaluated by its effective indicators.

The work done by gases in the cylinders of an engine is called indicator work. The indicator work of gases in one cylinder in one cycle is called cycle work. It can be determined using an indicator diagram built according to the data of the thermal calculation of the engine.

Area bounded by contour a -c-z"-z-b-a calculated indicator diagram A T , will represent, on an appropriate scale, the theoretical indicator work of gases in one cylinder per cycle. Real diagram area a"-c"-c"-z"-b"-b"-r-a-a" will consist of top and bottom loops. Square A d the upper loop characterizes the positive work of gases per cycle. The boundaries of this loop do not coincide with the calculated ones due to the ignition advance or fuel injection (s "-s- s"-s"), non-instantaneous combustion of fuel (with "-z" -z"-c" and z"- z-z""-z") and prerelease (b"-b-b"-b").

The reduction in the area of ​​the calculation diagram for the indicated reasons is taken into account using diagram completeness factor :

For automotive engines the values ​​of the completeness factor of the diagram take the values 0,93...0,97.

Square An the lower loop characterizes negative work spent on pumping passages piston for gas exchange in the cylinder. Thus, the actual indicator work of gases in one cylinder in one cycle:

In practice, the value of engine performance per cycle is determined by the average indicated pressure Pi, equal to the useful work of the cycle, referred to the unit of the working volume of the cylinder

Where Wi- useful work of the cycle, J (N m); Vh– working volume of the cylinder, m3.

Average indicator pressure - this is a conditionally constant pressure on the piston during one stroke of the piston, which does work equal to the indicator work of gases for the entire cycle. This pressure is expressed on a certain scale by the height pi rectangle with area A = Hell - An and with a base equal to the length of the indicator chart. Value pi during normal operation of the engine reaches, gasoline engines 1.2 MPa, in diesel engines - 1.0 MPa.

The useful work done by the gases in the engine cylinders per unit time is called the indicator power and denoted Pi .
The indicator work of gases in one cylinder for one cycle is (Nm)

Distinguish between average and true heat capacity. The average heat capacity cn is the amount of heat that is consumed when a unit of gas (1 kg, 1 m3, 1 mol) is heated by 1 K from t1 to t2:
с=q/(t2-t1)
The smaller the temperature difference t2 – t1, the more value the average heat capacity approaches the true s. Therefore, the true heat capacity will take place when the value of t2 - t1 approaches zero.

Heat capacity is a function of state parameters - pressure and temperature, therefore, in technical thermodynamics, true and average heat capacities are distinguished.

The heat capacity of an ideal gas depends only on temperature and, by definition, can only be found in the temperature range. However, it can always be assumed that this interval is very small near some temperature value. Then we can say that the heat capacity is determined at a given temperature. This heat capacity is called true.

In the reference literature, the dependence of the true heat capacities with p And with v temperature is given in the form of tables and analytical dependencies. An analytical dependence (for example, for mass heat capacity) is usually represented as a polynomial:

Then the amount of heat supplied in the process in the temperature range [ t1,t2] is determined by the integral:

In the study of thermodynamic processes, the average value of the heat capacity in the temperature range is often determined. It is the ratio of the amount of heat supplied in the process Q 12 to the final temperature difference:

Then, if the dependence of the true heat capacity on temperature is given, in accordance with (2):

Often in the reference literature, values ​​​​of the average heat capacities are given with p And with v for the temperature range from 0 before t about C. Like true ones, they are presented in the form of tables and functions:

When substituting the temperature value t this formula will be used to find the average heat capacity in the temperature range [ 0.t]. To find the average heat capacity in an arbitrary interval [ t1,t2], using dependence (4), it is necessary to find the amount of heat Q 12 applied to the system in this temperature range. Based on the rule known from mathematics, the integral in equation (2) can be divided into the following integrals:

After that, the desired value of the average heat capacity is found by formula (3).

Goal of the work

Experimentally determine the values ​​of the average heat capacity of air in the temperature range from t 1 to t 2, establish the dependence of the heat capacity of air on temperature.

1. Determine the power spent on gas heating from t 1

before t 2 .

2. Fix the values ​​of air flow in a given time interval.

Lab Preparation Guidelines

1. Work through the section of the course “Heat capacity” according to the recommended literature.

2. Familiarize yourself with this methodological guide.

3. Prepare protocols laboratory work, including the necessary theoretical material related to this work (calculation formulas, diagrams, graphs).

Theoretical introduction

Heat capacity- the most important thermophysical quantity, which is directly or indirectly included in all heat engineering calculations.

Heat capacity characterizes the thermophysical properties of a substance and depends on the molecular weight of the gas μ , temperature t, pressure R, the number of degrees of freedom of the molecule i, from the process in which heat is supplied or removed p = const, v =const. The heat capacity depends most significantly on the molecular weight of the gas μ . So, for example, the heat capacity for some gases and solids is

So the less μ , the less substance is contained in one kilomol and the more heat must be supplied to change the temperature of the gas by 1 K. That is why hydrogen is a more efficient coolant than, for example, air.

Numerically, heat capacity is defined as the amount of heat that must be brought to 1 kg(or 1 m 3), a substance to change its temperature by 1 K.

Since the amount of heat supplied dq depends on the nature of the process, then the heat capacity also depends on the nature of the process. The same system in different thermodynamic processes has different heat capacities - cp, cv, c n. Greatest practical value have cp And cv.

According to the molecular-kinematic theory of gases (MKT), for a given process, the heat capacity depends only on the molecular weight. For example, heat capacity cp And cv can be defined as

For air ( k = 1,4; R = 0,287 kJ/(kg· TO))

kJ/kg

For a given ideal gas, the heat capacity depends only on temperature, i.e.

The heat capacity of the body in this process called the ratio of heat dq obtained by the body with an infinitesimal change in its state to a change in body temperature by dt

True and average heat capacity

Under the true heat capacity of the working fluid is understood:

The true heat capacity expresses the value of the heat capacity of the working fluid at a point for given parameters.

The amount of transferred heat. expressed through the true heat capacity, can be calculated by the equation

Distinguish:

Linear dependence of heat capacity on temperature

Where A- heat capacity at t= 0 °С;

b = tgα - slope factor.

Nonlinear dependence of heat capacity on temperature.

For example, for oxygen, the equation is written as

kJ/(kg K)

Under medium heat capacity with t understand the ratio of the amount of heat in process 1-2 to the corresponding change in temperature

kJ/(kg K)

The average heat capacity is calculated as:

Where t = t 1 + t 2 .

Calculation of heat according to the equation

difficult, since the tables give the value of heat capacity. Therefore, the heat capacity in the range from t 1 to t 2 must be determined by the formula

.

If the temperature t 1 and t 2 is determined experimentally, then for m kg gas, the amount of heat transferred should be calculated according to the equation

Medium with t And With the true heat capacities are related by the equation:

For most gases, the higher the temperature t, the higher the heat capacity with v , with p. Physically, this means that the hotter the gas, the more difficult it is to heat it further.

Heat capacity is the ratio of the amount of heat imparted to the system to the temperature increase observed in this case (in the absence of chemical reaction, the transition of a substance from one state of aggregation to another and at A " = 0.)

Heat capacity is usually calculated per 1 g of mass, then it is called specific (J / g * K), or per 1 mol (J / mol * K), then it is called molar.

Distinguish average and true heat capacity.

Middle heat capacity is the heat capacity in the temperature range, i.e. the ratio of the heat imparted to the body to the increment in its temperature by ΔТ

True The heat capacity of a body is the ratio of an infinitesimal amount of heat received by the body to the corresponding increase in its temperature.

It is easy to establish a connection between the average and true heat capacity:

substituting the values ​​of Q into the expression for the average heat capacity, we have:

The true heat capacity depends on the nature of the substance, the temperature and the conditions under which the heat transfer to the system occurs.

So, if the system is enclosed in a constant volume, i.e. for isochoric process we have:

If the system expands or contracts while the pressure remains constant, i.e. For isobaric process we have:

But ΔQ V = dU, and ΔQ P = dH, therefore

C V = (∂U/∂T) v , and C P = (∂H/∂T) p

(if one or more variables are held constant while others change, then the derivatives are said to be partial with respect to the changing variable).

Both ratios are valid for any substances and any states of aggregation. To show the relationship between C V and C P, it is necessary to differentiate the expression for the enthalpy H \u003d U + pV /

For an ideal gas pV=nRT

for one mole or

The difference R is the work of the isobaric expansion of 1 mole of an ideal gas as the temperature rises by one unit.

For liquids and solids due to a small change in volume during heating С P = С V

Dependence of the thermal effect of a chemical reaction on temperature, Kirchhoff's equations.

Using Hess's law, one can calculate the thermal effect of the reaction at the temperature (usually 298 K) at which the measured standard heats formation or combustion of all participants in the reaction.

But more often it is necessary to know the thermal effect of a reaction at different temperatures.

Consider the reaction:

ν A A+ν B B= ν C С+ν D D

Let us denote by H the enthalpy of the participant in the reaction per 1 mole. The total change in the enthalpy ΔΗ (T) of the reaction is expressed by the equation:

ΔΗ \u003d (ν C H C + ν D H D) - (ν A H A + ν B H B); va, vb, vc, vd - stoichiometric coefficients. x.r.

If the reaction proceeds at constant pressure, then the change in enthalpy will be equal to the heat effect of the reaction. And if we differentiate this equation with respect to temperature, we get:

Equations for isobaric and isochoric process

And

called Kirchhoff equations(in differential form). They allow qualitatively evaluate the dependence of the thermal effect on temperature.

The influence of temperature on the thermal effect is determined by the sign of the value ΔС p (or ΔС V)

At ∆С p > 0 value , that is, with increasing temperature thermal effect increases

at ∆С p< 0 that is, as the temperature increases, the thermal effect decreases.

at ∆С p = 0- thermal effect of the reaction does not depend on temperature

That is, as follows from this, ΔС p determines the sign in front of ΔН.

This is the amount of heat that must be reported to the system to increase its temperature by 1 ( TO) Without useful work and the constancy of the corresponding parameters.

If we take an individual substance as a system, then total heat capacity of the system equals the heat capacity of 1 mole of a substance () times the number of moles ().

Heat capacity can be specific and molar.

Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by 1 hail(intense value).

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by 1 hail.

Distinguish between true and average heat capacity.

In engineering, the concept of average heat capacity is usually used.

Medium is the heat capacity for a certain temperature range.

If a system containing an amount of a substance or a mass was told the amount of heat , and the temperature of the system increased from to , then you can calculate the average specific or molar heat capacity:

True molar heat capacity- this is the ratio of an infinitesimal amount of heat imparted by 1 mole of a substance at a certain temperature to the temperature increase that is observed in this case.

According to equation (19), heat capacity, like heat, is not a state function. At constant pressure or volume, according to equations (11) and (12), heat, and, consequently, heat capacity acquire the properties of a state function, that is, they become characteristic functions of the system. Thus, we obtain isochoric and isobaric heat capacities.

Isochoric heat capacity- the amount of heat that must be reported to the system in order to increase the temperature by 1 if the process occurs at .

Isobaric heat capacity- the amount of heat that must be reported to the system in order to increase the temperature by 1 at .

The heat capacity depends not only on temperature, but also on the volume of the system, since there are interaction forces between particles that change with a change in the distance between them, therefore partial derivatives are used in equations (20) and (21).

The enthalpy of an ideal gas, like its internal energy, is a function of temperature only:

and in accordance with the Mendeleev-Clapeyron equation, then

Therefore, for an ideal gas in equations (20), (21), partial derivatives can be replaced by total differentials:

From the joint solution of equations (23) and (24), taking into account (22), we obtain the equation of the relationship between and for an ideal gas.

By dividing the variables in equations (23) and (24), we can calculate the change internal energy and enthalpy when 1 mol of ideal gas is heated from temperature to

If the heat capacity can be considered constant in the indicated temperature range, then as a result of integration we obtain:

Let us establish the relationship between the average and true heat capacity. The change in entropy, on the one hand, is expressed by equation (27), on the other hand,

Equating the right parts of the equations and expressing the average heat capacity, we have:

A similar expression can be obtained for the average isochoric heat capacity.

The heat capacity of most solid, liquid and gaseous substances increases with increasing temperature. The dependence of the heat capacity of solid, liquid and gaseous substances on temperature is expressed by an empirical equation of the form:

Where A, b, c and - empirical coefficients calculated on the basis of experimental data about , and the coefficient refers to organic matter, and - to inorganic. The values ​​of the coefficients for various substances are given in the handbook and are applicable only for specified interval temperatures.

The heat capacity of an ideal gas does not depend on temperature. According to the molecular kinetic theory, the heat capacity per one degree of freedom is equal to (the degree of freedom is the number of independent types of motion into which the complex motion of a molecule can be decomposed). A monatomic molecule is characterized by translational motion, which can be decomposed into three components in accordance with three mutually perpendicular directions along three axes. Therefore, the isochoric heat capacity of a monatomic ideal gas is

Then the isobaric heat capacity of a monatomic ideal gas according to (25) is determined by the equation

Diatomic molecules of an ideal gas, in addition to three degrees of freedom of translational motion, also have 2 degrees of freedom of rotational motion. Hence.