calculate the final temperature whenHelium (degree of freedom=5/3) at 1 atm pressure is comressed quasi-statically and adiabatically to a pressure of 5 atm.
To calculate the final temperature, we can use the adiabatic process equation:
\((\frac{T_2}{T_1}) = (\frac{P_2}{P_1})^{(\gamma-1)}\)
where:
\(T_1\) = Initial temperature
\(T_2\) = Final temperature
\(P_1\) = Initial pressure
\(P_2\) = Final pressure
\(\gamma\) = Heat capacity ratio
In this case, we have:
\(P_1 = 1\) atm
\(P_2 = 5\) atm
\(\gamma = \frac{5}{3}\) (degrees of freedom)
We can rearrange the formula to solve for \(T_2\):
\(T_2 = T_1 \cdot (\frac{P_2}{P_1})^{(\gamma-1)}\)
Since the process is adiabatic, there is no heat transfer (\(q = 0\)), so the initial and final temperatures are related by the equation:
\(T_2 = T_1 \cdot (\frac{P_2}{P_1})^{(\gamma-1)}\)
This equation relates the initial and final temperatures in terms of the pressure ratio and the heat capacity ratio.
To solve the problem, you need to know the initial temperature \(T_1\) and substitute the given values for \(P_1\), \(P_2\), and \(\gamma\) into the equation.