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Step 1: Identify the known information
• The gas is rigid and diatomic, so for 1 mole, the total degrees of freedom f = 5 .
• Heat supplied to the gas is Q .
• The work done by the gas is \frac{Q}{5} .
• We want to find the molar heat capacity C and compare it with the form \frac{xR}{8} to determine x .
Step 2: Apply the First Law of Thermodynamics
The First Law states:
Q = \Delta U + W.
Here, W = \frac{Q}{5}. Plugging this in:
Q = \Delta U + \frac{Q}{5} \quad \Longrightarrow \quad \Delta U = Q - \frac{Q}{5} = \frac{4Q}{5}.
Step 3: Express \Delta U in terms of C_v and temperature change
For 1 mole of a gas, the internal energy change is given by:
\Delta U = n C_v \Delta T.
Since n = 1 (1 mole):
\Delta U = C_v \Delta T.
We have already found \Delta U = \frac{4Q}{5}. So:
C_v \Delta T = \frac{4Q}{5}.
Step 4: Relate supplied heat Q to C \Delta T
The given process has a certain molar heat capacity C such that for 1 mole,
Q = C \Delta T.
Hence,
C \Delta T = Q.
Step 5: Find the ratio of C_v to C
Divide the expression for C_v \Delta T by the expression for C \Delta T :
\frac{C_v \Delta T}{C \Delta T} = \frac{\frac{4Q}{5}}{Q} = \frac{4}{5},
which gives
\frac{C_v}{C} = \frac{4}{5} \quad \Longrightarrow \quad C = \frac{5}{4} C_v.
Step 6: Substitute C_v for a rigid diatomic gas
For a diatomic gas (without vibrational degrees of freedom active),
C_v = \frac{f}{2} R = \frac{5}{2} R.
Thus,
C = \frac{5}{4} \times \frac{5}{2} R = \frac{25}{8} R.
Step 7: Compare with the given form to find x
We are told:
C = \frac{x R}{8}.
Comparing this with C = \frac{25}{8} R, we get x = 25.
Answer
The value of x is 25.
