For an ideal gas doing work in an adiabatic process, why is dU calculated by:
$$dU = C_v dT$$
The volume definitely must change in order for work to be done, so why is the specific heat that is used the one that assumes a constant volume?
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3 Answers
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Internal energy of an ideal gas depends only on the temperature of the gas.
Lets suppose, internal energy is $U_1$ at $T_1$ and $U_2$ at $T_2$,then $(U_1 -U_2)$ will only depend on $(T_1-T_2)$. The system can go from $U_1$ to $U_2$ by any process possible but you don't need to tell me what the process was,just tell me $T_1$ and $T_2$ and I will tell you what $(U_1 -U_2)$ will be.
Example :
For a monoatomic gas , $$U_1= \frac32 R T_1\,.$$ And, $$U_2 = \frac32 R T_2\,.$$ So, $$U_1-U_2 = 3/2 R (T_1-T_2)$$
Now in a constant volume process from first law and definition of $C_v$ we can say that,
$$dU = C_v dT$$ The change in internal energy for a temperature change $dT$ is $C_v dT$ for a constant volume process,but as I said already the process does not matter ,for a given change in temperature the change in internal energy will be same ,no matter what the process is. So why don't we use the formula $dU = C_v dT$ for an adiabatic process also ? Ofcourse we can.
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This is a different definition of change in internal energy and is calculated by keeping the volume constant.
Although in adiabatic process volume has to increase or decrease in order to do some work but this definition of ∆U in terms of specific heat is intact and independent of the change in volume of a system.
In adiabatic process, say we want to find the value of ∆U, so in order to do that we find the change in temperature of the system. Then we can use a separate system in which volume in constant and use this value of ∆T and get the value of ∆U (by using the heat capacity at constant volume formula) of the initial system under observation.
This is how it made sense to me..I had the same question..I hope this clears your problem..
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I think there are three points that deserve to be emphasized:
- The internal energy is a state function, that is the difference in the internal energies of two thermodynamic states depends only on the states. It cannot depend on the process going from one state to the other.
- The internal energy of the ideal gas depends only on the temperature.
Then not only for adiabatic process, but for any process the internal energy change is given by $dU=C_vdT$. To show this, choose two arbitrary points in a $pV$-diagram and join then by a composite path formed by an isochoric and an isothermal. By the first law and by the result (2) above we get $\Delta U=C_v \Delta T$. For infinitesimally different temperatures, $dU=C_vdT$. By (1), this result must be independent of the process.
The beauty of any state function (such as internal energy, potential energy, entropy...) is that we can compute it by choosing a convenient path/process.
Last but not least,
- Work is not always given by $W=\int pdV$.
The above expression is true only for reversible process. It is possible to do work on the system without changing its volume, for example by stirring an incompressible fluid or by doing electrical or magnetic works.