Heat capacity is a thermophysical characteristic that determines the ability of bodies to give or receive heat in order to change 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 a body (system of bodies): C=dQ/dT, where is the elementary amount of heat; - elementary temperature change.
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 will be J/K.
Depending on the quantitative unit of the body to which heat is supplied in thermodynamics, mass, volumetric and molar heat capacities are distinguished.
Mass heat capacity is the heat capacity per unit mass of the working fluid, c=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 is the heat capacity related to the amount of 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 relationship:
The volumetric heat capacity of gases is expressed in terms of molar heat capacity as
Where m 3 /mol is the molar volume of the gas under normal conditions.
Mayer's equation: C p – C v = R.
Considering that 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 they want to emphasize the dependence of the heat capacity of the working fluid on temperature, then they write it as C(t), and the specific heat capacity as c(t). Typically, true heat capacity is understood as the ratio of the elementary amount of heat that is imparted to a thermodynamic system in any process to the infinitesimal increase in the temperature of this system caused by the imparted heat. We will consider C(t) to be the true heat capacity of the thermodynamic system at a system temperature equal to t 1 , and c(t) to be the true specific heat capacity of the working fluid at its temperature equal to t 2 . Then the average specific heat capacity of the working fluid when its temperature changes from t 1 to t 2 can be determined as
Typically, the tables give average values of heat capacity c av for various temperature intervals starting with t 1 = 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 tabulated values of average heat capacities c av as follows.
The perfection of thermal processes occurring in the cylinder of a real automobile engine is assessed 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 assessed by its effective indicators.
The work done by gases in the engine cylinders 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 based on engine thermal calculation data
Area bounded by contour a -c-z"-z-b-a calculated indicator chart A T , will, on an appropriate scale, represent the theoretical indicator work of gases in one cylinder per cycle. Area of a real diagram 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 ignition timing or fuel injection (с"-с- s"-s"), non-instant fuel combustion (with "-z" -z"-с" and z"- z-z""-z") and release prefixes (b"-b-b"-b").
The reduction in the area of the calculation diagram for the specified reasons is taken into account using diagram completeness factor :
For automobile and tractor engines the values of the diagram completeness coefficient take values 0,93...0,97.
Square An the bottom loop characterizes negative work spent on pump strokes piston for gas exchange in the cylinder. Thus, the actual indicator work of gases in one cylinder per cycle:
In practice, the amount of engine performance per cycle is determined by the average indicator pressure Pi, equal to the useful work of the cycle per unit of working volume of the cylinder
Where Wi- useful work of the cycle, J(N m); Vh– cylinder working volume, 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 height pi rectangle with area A = Hell - An and with a base equal to the length of the indicator diagram. Magnitude pi during normal engine operation it reaches gasoline engines 1.2 MPa, in diesel engines - 1.0 MPa.
The useful work done by gases in the engine cylinders per unit time is called indicator power and is denoted Pi
.
The indicator work of gases in one cylinder per cycle is (Nm)
Distinguish between average and true heat capacity. Average heat capacity c„ is the amount of heat that is consumed when heating a unit of gas (1 kg, 1 m3, 1 mol) 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 c. Consequently, the true heat capacity will occur when the value 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 be found only in the temperature range. However, we can always assume that this interval is very small near any 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 true heat capacities with p And with v on temperature are specified in the form of tables and analytical dependencies. The analytical relationship (for example, for mass heat capacity) is usually represented as a polynomial:
Then the amount of heat supplied during the process in the temperature range [ t1,t2] is determined by the integral:
When studying thermodynamic processes, the average heat capacity value over a 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 reference literature the values of average heat capacities are given with p And with v for the temperature range from 0 to t o C. Like true ones, they are represented in the form of tables and functions:
When substituting the temperature value t This formula will find the average heat capacity in the temperature range [ 0,t]. To find the average value of heat capacity in an arbitrary interval [ t1,t2], using relationship (4), you need to find the amount of heat Q 12, supplied to the system in this temperature range. Based on a rule known from mathematics, the integral in equation (2) can be divided into the following integrals:
After this, the desired value of the average heat capacity is found using formula (3).
Purpose of the work
To 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 heating gas from t 1
to t 2 .
2. Record the air flow values in a given time interval.
Instructions for preparing for laboratory work
1. Work through the section of the course “Heat capacity” using the recommended literature.
2. Familiarize yourself with this methodological manual.
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 thermotechnical calculations.
Heat capacity characterizes the thermophysical properties of a substance and depends on the molecular weight of the gas μ , temperature t, pressure r, 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 most significantly depends on the molecular weight of the gas μ . For example, the heat capacity for some gases and solids is
Thus, the less μ , the less substance is contained in one kilomole and the more heat must be supplied to change the temperature of the gas by 1 K. This is why hydrogen is a more effective 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), substances 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 - c p, c v, c n. Greatest practical significance have c p And c v.
According to the molecular kinematic theory of gases (MKT), for a given process the heat capacity depends only on the molecular mass. For example, heat capacity c p And c v 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 heat ratio dq, obtained by a body with an infinitesimal change in its state to a change in body temperature by dt
True and average heat capacities
The true heat capacity of the working fluid is understood as:
True heat capacity expresses the value of the heat capacity of the working fluid at a point with given parameters.
The amount of heat transferred. expressed in terms of true heat capacity, can be calculated using the equation
There are:
Linear dependence of heat capacity on temperature
Where A- heat capacity at t= 0 °C;
b = tgα - angular coefficient.
Nonlinear dependence of heat capacity on temperature.
For example, for oxygen the equation is represented as
kJ/(kg K)
Below average heat capacity with t understand the ratio of the amount of heat in process 1-2 to the corresponding temperature change
kJ/(kg K)
The average heat capacity is calculated as:
Where t = t 1 + t 2 .
Calculation of heat using 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 using the equation
Average with t And With true heat capacity is 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 is, 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 observed temperature increase (in the absence 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.
Average heat capacity is the heat capacity in the temperature range, i.e. the ratio of the heat imparted to the body to the increase in its temperature by the value ΔT
True The heat capacity of a body is the ratio of the 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:
True heat capacity depends on the nature of the substance, temperature and conditions under which 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, but 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 vary, then the derivatives are said to be partial with respect to the changing variable).
Both relationships are valid for any substance and any state of aggregation. To show the connection between C V and C P, it is necessary to differentiate by temperature the expression for enthalpy H = U + pV /
For an ideal gas pV=nRT
for one mole or
The difference R represents the work of isobaric expansion of 1 mole of an ideal gas as the temperature increases by one unit.
In liquids and solids due to the small change in volume upon heating C P = C V
Dependence of the thermal effect of a chemical reaction on temperature, Kirchhoff equations.
Using Hess's law, it is possible to calculate the thermal effect of the reaction at the temperature (usually 298K) at which the standard heats formation or combustion of all reaction participants.
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 C+ν D D
Let us denote by H the enthalpy of a reaction participant per 1 mole. The total change in enthalpy ΔΗ(T) of the reaction will be expressed by the equation:
ΔΗ = (ν C Н С +ν D Н D) - (ν A Н А +ν B Н В); va, vb, vc, vd - stoichiometric coefficients. h.r.
If the reaction proceeds at constant pressure, then the change in enthalpy will be equal to the thermal effect of the reaction. And if we differentiate this equation by temperature, we get:
Equations for isobaric and isochoric processes
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 ΔС p > 0 value, that is, with increasing temperature the thermal effect increases
at ΔС p< 0 that is, as the temperature increases, the thermal effect decreases.
at ΔС p = 0- thermal effect of the reaction independent of 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 supplied to the system to increase its temperature by 1 ( TO) in the absence 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 () multiplied by the number of moles ().
Heat capacity can be specific and molar.
Specific heat is the amount of heat required to heat a unit mass of a substance by 1 hail(intensive value).
Molar heat capacity is the amount of heat required to heat one mole of a substance by 1 hail.
There are true and average heat capacity.
In engineering, the concept of average heat capacity is usually used.
Average is the heat capacity for a certain temperature range.
If a system containing an amount of substance or mass is given an amount of heat, and the temperature of the system increases from to, then the average specific or molar heat capacity can be calculated:
True molar heat capacity- this is the ratio of the infinitesimal amount of heat imparted by 1 mole of a substance at a certain temperature to the temperature increment that is observed at the same time.
According to equation (19), heat capacity, like heat, is not a function of state. At constant pressure or volume, according to equations (11) and (12), heat, and, consequently, heat capacity acquires 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 supplied to the system to increase the temperature by 1, if the process occurs at.
Isobaric heat capacity- the amount of heat that must be supplied to the system to increase the temperature by 1 at .
Heat capacity depends not only on temperature, but also on the volume of the system, since there are interaction forces between particles that change when the distance between them changes, 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 an equation for 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 heating 1 mole of an ideal gas from a temperature to
If the heat capacity can be considered constant in the specified 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-hand sides 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 solids, liquids and gases increases with 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 are empirical coefficients calculated on the basis of experimental data on , and the coefficient refers to organic matter, and - to inorganic. The coefficient values for various substances are given in the reference book 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 degree of freedom is equal to (degree of freedom - 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 equal to
Then the isobaric heat capacity of a monatomic ideal gas, according to (25), will be 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.
