[VW, S& B: 5.6]
2.4 Specific Heats: the relation between temperature change and heat
How much does a given amount of heat transfer change the temperature of a substance? It depends on the substance. In general
where is a constant that depends on the substance.
We ремон моторо чери истар 2,4 determine the constant for any substance if we know how much heat is transferred. Since heat is path dependent, however, we must specify the ремон моторо чери истар 2,4, i.e., the path, to find .
Two useful processes are constant pressure and constant volume, so we will consider these each in turn. We will call the specific heat at constant pressure and that at constant volume or and per unit mass.
- The Specific Heat at Constant Ремон моторо чери истар 2,4
Remember that if we specify any two properties of the system, then the state of the system is fully specified. In other words we can write or .
[VW, S & B: ремон моторо чери истар 2,4
Consider the form and use the chain rule to write how changes with respect to and :
For a constant volume process, the second term is zero since there is no change in volume, .
Now if we write the First Law for a quasi-static process, with
we see that again the second term is zero if the process is also constant volume.
Equating (2.5) and (2.6) with canceled in each,
In this case, any energy increase is due only to energy transfer as heat.
We can therefore use our definition of specific heat from Equation (2.4) to define the specific heat for a constant volume process,
- The Specific Heat at Constant Pressure
If we write and consider a constant pressure process, we can perform a similar derivation to the one above and show that
In the derivation of ремон моторо чери истар 2,4 src="http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/img157.png">we considered only a constant volume process, hence the name, ``specific heat at constant volume.'' It is more useful, however, to think of in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than as a quantity related to heat transfer in a special process.
Ремон моторо чери истар 2,4 fact, the derivatives above are defined at any point in any quasi-static process whether that process is constant volume, constant pressure, or neither.
The names ``specific heat at constant volume'' and ``specific heat at constant pressure'' are therefore unfortunate misnomers; and are thermodynamic properties of a substance, and by definition depend only the state.
They are extremely important values, and have been experimentally determined as a function of ремон моторо чери истар 2,4 thermodynamic state for an enormous number of simple compressible substances2.1.
Practice Questions Throw an object from the top tier of the lecture hall to the front of the room.
Estimate how much the temperature of the room has changed as a result. Start by listing what information you need to solve this problem.
2.4.1 Specific Heats of an Ideal Gas
The equation of state for an ideal gas is
Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a ремон моторо чери истар 2,4 as an ideal gas constitutes a model of the actual physical situation, and the limits of model validity must always be kept in mind.
One of the other important features of an ideal gas ремон моторо чери истар 2,4 that its internal energy depends only upon its temperature. (For now, this can be regarded as another aspect of the model of actual systems that the ideal gas represents, but it can be shown that this is a consequence of the form of the equation of state.) Since depends only on ,
In the above equation we have indicated that can depend on .
Like the internal energy, the enthalpy is also only dependent on for an ideal gas.
(If is a function of then, using the ideal gas equation of state, is also.) Therefore,
If we are interested in finite changes of internal energy or enthalpy, we integrate,
Over small temperature changes ( ), it is often assumed that and are constant.
Furthermore, there are wide ranges over which specific heats do not vary greatly with respect to temperature, as shown in SB&VW Figure 5.11. It is thus often useful to treat them as constant. If so
These equations are useful in ремон моторо чери истар 2,4 internal energy or enthalpy differences, but it should be remembered that they hold only if the specific heats are constant.
We can relate the specific heats ремон моторо чери истар 2,4 an ideal gas to its gas constant as follows. We write the first law in terms of internal energy,
and assume a quasi-static process so that we can also write it in terms of enthalpy, as in Section 2.3.4,
Equating the two first law expressions above, and assuming an ideal gas, we obtain
An expression that will appear often is the ratio of specific heats, which ремон моторо чери истар 2,4 will define as
Below we summarize the important results for all ideal gases, and ремон моторо чери истар 2,4 some values for specific types of ideal gases.
- All ideal gases:
- The specific heat at constant volume ( for a unit mass or for one kmol) is a function of only.
- The specific heat at constant pressure ( for a unit mass or for one kmol) is a function of only.
- A relation that connects the specific heats and the gas constant is where the units depend on the mass considered.
For a unit mass of gas, e.g., a kilogram, and would be the specific heats for one kilogram of gas and is as defined above.
For one kmol of gas, the expression takes the formwhere and have been used to denote the specific heats for one kmol of gas and is the universal gas constant.
- The specific heat ratio, (or ), is a function of only and is greater than unity.
- An ideal gas with specific ремон моторо чери истар 2,4 independent of temperature, and is referred to as a perfect gas.
For example, monatomic gases and diatomic gases at ordinary temperatures are considered perfect gases. To make this distinction the terminology "a perfect gas with constant specific heats" is used throughout the notes. In some textbooks perfect gases are sometimes also referred to as ideal gases, and to avoid confusion we use the stated terminology2.2.
- Monatomic gases, such as He, Ne, Ar, and most metallic vapors:
- (or ) is constant over a wide temperature range and is very nearly equal to ремон моторо чери истар 2,4 [or for one kmol].
- (or ) is constant over a wide temperature range and is very nearly equal to [or for one kmol].
- is constant over a wide temperature range and is very nearly equal to [ ].
- So-called permanent diatomic gases, namely HON Air, NO, and CO:
- (or ) is nearly constant at ordinary temperatures, being approximately [ for one kmol], and increases slowly at higher temperatures.
- ремон моторо чери истар 2,4 (or ) is nearly constant ремон моторо чери истар 2,4 ordinary temperatures, being approximately [ for one kmol], and increases slowly at higher temperatures.
- is constant over a temperature range of roughly to ремон моторо чери истар 2,4 is very nearly equal to [ ].
It decreases with temperature above this.
- Polyatomic gases and gases that are chemically active, such as CONHCHand Freons:
The specific heats, and and vary with the temperature, the variation being different for each gas.
The general trend is that heavy molecular weight gases (i.e., more complex gas molecules than those listed in 2 or 3), have values of closer to unity than diatomic gases, which, as can be seen above, are closer to unity than monatomic gases.
For example, values of below 1.2 are typical of Freons which have molecular weights of over one hundred.2.3
In general, for substances other than ideal gases, and depend on pressure as well as on temperature, and the above relations will not all apply.
In this respect, the ideal gas is a very special model.
In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume.
The simple relations between changes in energy (or enthalpy) and temperature are a consequence of the behavior of an ideal gas, specifically the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances.2.4
2.4.2 Reversible adiabatic processes for an ideal gas
From the first law, with and
Also, using the definition of enthalpy,
The underlined terms are zero for an adiabatic process.
Rewriting (2.7) and (2.8),
Equation (2.9) can be integrated between states 1 and 2 to give
We can substitute for or in the above result using the ideal gas law, or carry out the derivation slightly differently, to also show that
We will use the above equations to relate pressure and temperature to one another for quasi-static adiabatic processes (for instance, this type of process is our idealization of what happens in compressors and turbines).
- On a - diagram for a closed-system sketch the thermodynamic paths that the system would follow if expanding from by isothermal and quasi-static, adiabatic processes.
- For which process is the most work done by the system?
- For which process is there heat exchange? Is it ремон моторо чери истар 2,4 or removed?
- Is the final state of the system the same after each process?
- Derive expressions for the work ремон моторо чери истар 2,4 by the system for each process. ремон моторо чери истар 2,4 UnifiedTP