In Mayer's relation:
`C_(P)-C_(V)=R`
'R' stands for:

Options

A. translational kinetic energy of 1 mol gas
B. rotational kinetic energy of 1 mol gas
C. vibrational kinetic energy of 1 mol gas
D. work done to increase the temperature of 1 mol gas by one degree

Answer

To solve the question regarding Mayer's relation \( C_{P} - C_{V} = R \), we need to analyze what \( R \) represents in this context. Here's a step-by-step breakdown of the solution:

### Step 1: Understand Mayer's Relation
Mayer's relation states that the difference between the heat capacities at constant pressure (\( C_{P} \)) and constant volume (\( C_{V} \)) for an ideal gas is equal to the gas constant \( R \):
\[ C_{P} - C_{V} = R \]

### Step 2: Identify the Variables
In this relation:
- \( C_{P} \) is the molar heat capacity at constant pressure.
- \( C_{V} \) is the molar heat capacity at constant volume.
- \( R \) is the universal gas constant.

### Step 3: Consider the Ideal Gas Law
The ideal gas law is given by:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]

### Step 4: Analyze Changes in Volume and Temperature
Now, consider a small change in volume (\( \Delta V \)) and an increase in temperature by 1 degree Celsius. The new state can be expressed as:
\[ PV + P\Delta V = R(T + 1) \]

### Step 5: Substitute and Rearrange
Substituting \( PV = RT \) into the equation gives:
\[ RT + P\Delta V = RT + R \]
By rearranging, we find:
\[ P\Delta V = R \]

### Step 6: Relate Work Done to \( R \)
The work done (\( W \)) during this process can be expressed as:
\[ W = P\Delta V \]
From our previous step, we see that:
\[ W = R \]

### Conclusion
Thus, we conclude that \( R \) represents the work done to increase the temperature of one mole of gas by 1 degree Celsius.

### Final Answer
In Mayer's relation, \( R \) stands for the work done to increase the temperature of 1 mole of gas by 1 degree Celsius.

---