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A thermodynamic process obeys the follow...

A thermodynamic process obeys the following relation `2dQ=dU+2dW` where dQ, dU, dW has ussual meaning then at which statement is true [given di-atomic gas, R=gas constant] then heat capacity for the process is

A

`(5R)/(2)`

B

`(7R)/(2)`

C

`(3R)/(5)`

D

infinite.

Text Solution

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The correct Answer is:
To solve the problem, we start with the given thermodynamic relation: \[ 2dQ = dU + 2dW \] ### Step 1: Rearranging the Equation We can rearrange this equation to express \( dQ \) in terms of \( dU \) and \( dW \). \[ 2dQ = dU + 2dW \] Dividing the entire equation by 2 gives us: \[ dQ = \frac{1}{2}dU + dW \] ### Step 2: Expressing \( dU \) From the first law of thermodynamics, we know that: \[ dQ = dU + dW \] Now, we can express \( dU \) in terms of \( dQ \) and \( dW \): \[ dU = dQ - dW \] ### Step 3: Substituting \( dU \) into the Rearranged Equation Now, substitute \( dU \) from the above equation into our rearranged equation: \[ dQ = \frac{1}{2}(dQ - dW) + dW \] ### Step 4: Simplifying the Equation Expanding the equation gives: \[ dQ = \frac{1}{2}dQ - \frac{1}{2}dW + dW \] Combining like terms results in: \[ dQ = \frac{1}{2}dQ + \frac{1}{2}dW \] ### Step 5: Isolating \( dQ \) To isolate \( dQ \), we can subtract \( \frac{1}{2}dQ \) from both sides: \[ dQ - \frac{1}{2}dQ = \frac{1}{2}dW \] This simplifies to: \[ \frac{1}{2}dQ = \frac{1}{2}dW \] Thus, we find: \[ dQ = dW \] ### Step 6: Analyzing the Result From the relation \( dQ = dW \), we can conclude that there is no change in internal energy (\( dU = 0 \)). This indicates that the process is isothermal (constant temperature) since the internal energy change for an ideal gas depends only on temperature. ### Step 7: Heat Capacity for the Process In an isothermal process, the heat capacity \( C \) is defined as: \[ C = \frac{dQ}{dT} \] Since \( dT = 0 \) in an isothermal process, the heat capacity becomes infinite: \[ C = \infty \] ### Conclusion Thus, the heat capacity for the process is infinite. ### Final Answer The heat capacity for the process is infinite. ---
To solve the problem, we start with the given thermodynamic relation: \[ 2dQ = dU + 2dW \] ### Step 1: Rearranging the Equation We can rearrange this equation to express \( dQ \) in terms of \( dU \) and \( dW \). \[ 2dQ = dU + 2dW \] ...
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