- #1
zenterix
- 629
- 82
- Homework Statement
- I have a quick question about a relatively simple problem.
Suppose we have 1 mole of a specified gas (say, helium) and it undergoes a reversible adiabatic process.
We are given initial volume ##V_1## and final volume ##V_2## and initial temperature ##T_1##.
- Relevant Equations
- If we want to calculate ##T_2## and ##P_2##, is is mandatory that we determine what ##\gamma=\frac{C_P}{C_V}## is?
This problem does not specify if helium is to be treated as an ideal gas. I am assuming it is an ideal gas.
We can determine ##P_1## from the ideal gas law
$$P_1=\frac{nRT_1}{V_1}\tag{1}$$
Since the process is adiabatic we have ##\delta q=0## and
##\dU=C_V dT=-PdV=-\frac{nRT}{V}dV\tag{2}##
After integrating and some algebra we reach
$$\frac{T_2}{T_1}=\left (\frac{V_1}{V_2}\right )^{\gamma-1}\tag{3}$$
From this we can sub in for temperature using the ideal gas law to reach
$$\left (\frac{V_1}{V_2}\right )^\gamma=\frac{P_2}{P_1}\tag{4}$$
$$PV^\gamma=\text{constant}=k\tag{5}$$
We don't know what ##k## because we don't know what ##\gamma is##.
If we knew gamma we would have ##k## from ##P_1V_1^\gamma## and we could then compute ##P_2##.
We could also then compute ##T_2## from (3).
To find ##\gamma## we can do
$$\gamma=\frac{C_P}{C_V}=\frac{C_P}{C_P-nR}$$
and so we just need to look up ##C_P## for helium at ##T_1##.
In the integration to reach (3) we already assumed that ##C_V## is constant which means ##C_P=C_V+nR## must be constant.
This is how I solved the problem.
I just want to make sure there isn't another way. Do we need to look up heat capacity to solve this problem in this way or is there another way?
We can determine ##P_1## from the ideal gas law
$$P_1=\frac{nRT_1}{V_1}\tag{1}$$
Since the process is adiabatic we have ##\delta q=0## and
##\dU=C_V dT=-PdV=-\frac{nRT}{V}dV\tag{2}##
After integrating and some algebra we reach
$$\frac{T_2}{T_1}=\left (\frac{V_1}{V_2}\right )^{\gamma-1}\tag{3}$$
From this we can sub in for temperature using the ideal gas law to reach
$$\left (\frac{V_1}{V_2}\right )^\gamma=\frac{P_2}{P_1}\tag{4}$$
$$PV^\gamma=\text{constant}=k\tag{5}$$
We don't know what ##k## because we don't know what ##\gamma is##.
If we knew gamma we would have ##k## from ##P_1V_1^\gamma## and we could then compute ##P_2##.
We could also then compute ##T_2## from (3).
To find ##\gamma## we can do
$$\gamma=\frac{C_P}{C_V}=\frac{C_P}{C_P-nR}$$
and so we just need to look up ##C_P## for helium at ##T_1##.
In the integration to reach (3) we already assumed that ##C_V## is constant which means ##C_P=C_V+nR## must be constant.
This is how I solved the problem.
I just want to make sure there isn't another way. Do we need to look up heat capacity to solve this problem in this way or is there another way?