A polyatomic ideal gas with linear structure is being supplied that Q in a polytropic process. The work done upon gas is `(Q)/(N)`. The molar specific heat of the gas for the process is `(N'NR)/(2[N+1])`. Then find the value of N' (here N is a constant).

Answer

To solve the problem step by step, we will use the first law of thermodynamics and the properties of a polyatomic ideal gas.

### Step 1: Apply the First Law of Thermodynamics
The first law of thermodynamics states that:
\[ \Delta Q = \Delta U + \Delta W \]
Where:
- \(\Delta Q\) is the heat added to the system,
- \(\Delta U\) is the change in internal energy,
- \(\Delta W\) is the work done by the system.

In this case, we are given:
- \(\Delta Q = Q\)
- The work done on the gas is given as \(\Delta W = -\frac{Q}{N}\) (since work done on the gas is negative).

Thus, we can write:
\[ Q = \Delta U - \frac{Q}{N} \]

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ Q + \frac{Q}{N} = \Delta U \]
Factoring out \(Q\):
\[ Q\left(1 + \frac{1}{N}\right) = \Delta U \]

### Step 3: Expressing Change in Internal Energy
For a polyatomic ideal gas with linear structure, the degrees of freedom \(F\) is given by:
\[ F = 5 \quad (\text{3 translational + 2 rotational}) \]
The change in internal energy \(\Delta U\) can be expressed as:
\[ \Delta U = \frac{F}{2} R T \]
Substituting \(F = 5\):
\[ \Delta U = \frac{5}{2} R T \]

### Step 4: Equating the Two Expressions for \(\Delta U\)
From Step 2, we have:
\[ Q\left(1 + \frac{1}{N}\right) = \frac{5}{2} R T \]
Now, we can express \(\frac{Q}{T}\):
\[ \frac{Q}{T} = \frac{5R}{2} \cdot \frac{1}{1 + \frac{1}{N}} = \frac{5R N}{2(N + 1)} \]

### Step 5: Relating to Molar Specific Heat
The molar specific heat for the process is given as:
\[ C = \frac{N' N R}{2(N + 1)} \]
Setting the two expressions for \(\frac{Q}{T}\) equal to each other:
\[ \frac{5R N}{2(N + 1)} = \frac{N' N R}{2(N + 1)} \]

### Step 6: Canceling Common Terms
Since \(R\) and \(\frac{1}{2(N + 1)}\) are common on both sides, we can cancel them:
\[ 5N = N' \]

### Step 7: Final Result
Thus, we find:
\[ N' = 5 \]

### Conclusion
The value of \(N'\) is \(5\).

---