Work done in an isobaric process is 100 Joules. If adiabatic constant for the gas is 1.4, find the heat given to the gas.
Show Hint
In an isobaric process, the heat supplied to the gas is equal to the work done plus the change in internal energy. For an ideal gas, the adiabatic constant \( \gamma = \frac{C_P}{C_V} \) relates the specific heats.
Step 1: Use the first law of thermodynamics.
The first law of thermodynamics states that the change in internal energy (\( \Delta U \)) is equal to the heat (\( Q \)) supplied to the system minus the work (\( W \)) done by the system:
\[
\Delta U = Q - W
\]
For an isobaric process (constant pressure), the work done is given by:
\[
W = P \Delta V
\]
where \( P \) is the pressure and \( \Delta V \) is the change in volume.
Step 2: Use the relationship for heat in an isobaric process.
The heat supplied in an isobaric process is related to the change in temperature by:
\[
Q = n C_P \Delta T
\]
where \( C_P \) is the specific heat at constant pressure and \( n \) is the number of moles of the gas.
Now, we know that:
\[
\Delta U = n C_V \Delta T
\]
and the relationship between \( C_P \) and \( C_V \) is:
\[
C_P = C_V + R
\]
where \( R \) is the universal gas constant.
Step 3: Apply the adiabatic constant.
For an ideal gas, the adiabatic constant \( \gamma \) is the ratio of the specific heats:
\[
\gamma = \frac{C_P}{C_V}
\]
Given that \( \gamma = 1.4 \), we can relate \( C_P \) and \( C_V \) to solve for the heat given to the gas.
The work done in an isobaric process is related to the heat supplied by:
\[
Q = W + \Delta U
\]
Substituting the values for \( W = 100 \, \text{J} \) and solving for \( Q \), we get:
\[
Q = 100 + 250 = 350 \, \text{J}
\]
Thus, the heat given to the gas is 350 J.
