The quantity of heat required to raise one mole through `1K` for a monoatomic gas at constant volume is

Options

A. `(3)/(2)R`
B. `(5)/(2)R`
C. `(7)/(2)R`
D. `4R`

Explanation

To solve the problem of finding the quantity of heat required to raise the temperature of one mole of a monoatomic gas by 1 K at constant volume, we can follow these steps:

### Step 1: Understand the concept of specific heat at constant volume (Cv)
The specific heat at constant volume (Cv) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin (or one degree Celsius) while keeping the volume constant.

### Step 2: Use the relationship between internal energy and specific heat
For a monoatomic ideal gas, the specific heat at constant volume (Cv) can be expressed in terms of the change in internal energy (U) with respect to temperature (T):
\[ C_v = \frac{dU}{dT} \]

### Step 3: Calculate the internal energy (U) for a monoatomic gas
The internal energy (U) for a monoatomic ideal gas is given by the formula:
\[ U = \frac{f}{2} nRT \]
where:
- \( f \) is the degrees of freedom (for monoatomic gas, \( f = 3 \)),
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

### Step 4: Substitute the values for a monoatomic gas
For one mole of a monoatomic gas (\( n = 1 \)):
\[ U = \frac{3}{2} RT \]

### Step 5: Differentiate U with respect to T
Now, we differentiate U with respect to T to find Cv:
\[ C_v = \frac{dU}{dT} = \frac{d}{dT} \left( \frac{3}{2} RT \right) \]
\[ C_v = \frac{3}{2} R \]

### Step 6: Conclusion
Thus, the quantity of heat required to raise the temperature of one mole of a monoatomic gas by 1 K at constant volume is:
\[ Q = C_v \cdot \Delta T = \frac{3}{2} R \cdot 1 = \frac{3}{2} R \]

### Final Answer:
The quantity of heat required is \( \frac{3}{2} R \).

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