The volume of one mole of ideal gas with adiabatic exponent is varied according to law `V = 1/T`. Find amount of heat obtained by gas in this process if gas temperature is increased by 100 K.

Options

A. `100 R`
B. `200 R`
C. `300 R`
D. `400 R`

Explanation

To solve the problem step by step, we will follow the given information and apply relevant physics concepts.

### Step 1: Understand the relationship given in the problem
The volume of the gas varies according to the law:
\[ V = \frac{1}{T} \]
This implies that:
\[ V \cdot T = 1 \]
This relationship indicates that the product of volume and temperature is a constant.

### Step 2: Use the Ideal Gas Law
From the ideal gas equation, we know:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]
We can rearrange this to express temperature \( T \):
\[ T = \frac{PV}{R} \]

### Step 3: Substitute \( T \) in the volume equation
Substituting \( T \) into the equation \( V \cdot T = 1 \):
\[ V \cdot \left(\frac{PV}{R}\right) = 1 \]
This leads to:
\[ \frac{PV^2}{R} = 1 \]
From this, we can derive:
\[ PV^2 = R \]
This shows that \( PV^2 \) is a constant.

### Step 4: Relate this to the adiabatic process
The equation \( PV^\gamma = \text{constant} \) is the form of the adiabatic process. Since we have \( PV^2 = \text{constant} \), we can equate:
\[ \gamma = 2 \]

### Step 5: Calculate the molar specific heat \( C \)
The molar specific heat \( C \) can be calculated using:
\[ C = C_v + \frac{R}{1 - \gamma} \]
For a monatomic ideal gas, the molar specific heat at constant volume \( C_v \) is:
\[ C_v = \frac{3}{2} R \]
Substituting \( \gamma = 2 \):
\[ C = \frac{3}{2} R + \frac{R}{1 - 2} \]
This simplifies to:
\[ C = \frac{3}{2} R - R = \frac{1}{2} R \]

### Step 6: Calculate the heat obtained by the gas
The amount of heat \( Q \) obtained by the gas when the temperature increases by \( \Delta T = 100 \, K \) is given by:
\[ Q = nC\Delta T \]
Substituting the values:
\[ Q = 1 \cdot \frac{1}{2} R \cdot 100 \]
Thus:
\[ Q = 50R \]

### Final Answer
The amount of heat obtained by the gas in this process is:
\[ Q = 50R \]

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