If R = universal gas constant, the amount of heat needed to raise the temperature of 2 mole of an ideal monoatomic gas from 273 K to 373 K when no work is done

Options

A. 100 R
B. 150 R
C. 300 R
D. 500R

Answer

To solve the problem of calculating the amount of heat needed to raise the temperature of 2 moles of an ideal monoatomic gas from 273 K to 373 K when no work is done, we will follow these steps:

### Step 1: Identify the change in temperature (ΔT)
The initial temperature (T1) is 273 K and the final temperature (T2) is 373 K.

\[ \Delta T = T2 - T1 = 373 \, \text{K} - 273 \, \text{K} = 100 \, \text{K} \]

### Step 2: Determine the number of moles (N)
The problem states that we have 2 moles of gas.

\[ N = 2 \, \text{moles} \]

### Step 3: Use the formula for change in internal energy (ΔU)
For a monoatomic ideal gas, the change in internal energy (ΔU) can be calculated using the formula:

\[ \Delta U = \frac{3}{2} N R \Delta T \]

Where:
- \(R\) is the universal gas constant (approximately 8.314 J/(mol·K)).
- \(N\) is the number of moles.
- \(\Delta T\) is the change in temperature.

### Step 4: Substitute the values into the formula
Substituting the values we have:

\[ \Delta U = \frac{3}{2} \times 2 \, \text{moles} \times R \times 100 \, \text{K} \]

\[ \Delta U = 3 \times R \times 100 \]

\[ \Delta U = 300 R \]

### Step 5: Apply the first law of thermodynamics
According to the first law of thermodynamics:

\[ \Delta Q = \Delta U + \Delta W \]

Since no work is done (ΔW = 0), we have:

\[ \Delta Q = \Delta U \]

### Step 6: Conclusion
Thus, the amount of heat needed to raise the temperature of the gas is:

\[ \Delta Q = 300 R \]

### Final Answer:
The amount of heat needed is \(300 R\).

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