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

Options

A. a. 100R
B. b. 150R
C. c. 300R
D. d. 500R

Explanation

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

### Step 1: Understand the relationship between heat and internal energy
In thermodynamics, the heat added to a system (ΔQ) is related to the change in internal energy (ΔU) and the work done (W) on the system by the equation:
\[ \Delta Q = \Delta U + W \]
Since the problem states that no work is done (W = 0), we can simplify this to:
\[ \Delta Q = \Delta U \]

### Step 2: Determine the change in internal energy (ΔU)
The change in internal energy for an ideal gas can be calculated using the formula:
\[ \Delta U = n C_v \Delta T \]
where:
- \( n \) = number of moles of gas
- \( C_v \) = molar heat capacity at constant volume
- \( \Delta T \) = change in temperature

### Step 3: Identify the values for the variables
From the problem:
- \( n = 2 \) moles (given)
- For a monatomic ideal gas, \( C_v = \frac{3R}{2} \)
- The initial temperature \( T_1 = 273 \, K \) and the final temperature \( T_2 = 373 \, K \)

Now, calculate the change in temperature:
\[ \Delta T = T_2 - T_1 = 373 \, K - 273 \, K = 100 \, K \]

### Step 4: Substitute the values into the ΔU equation
Now we can substitute the values into the equation for ΔU:
\[ \Delta U = n C_v \Delta T \]
\[ \Delta U = 2 \left(\frac{3R}{2}\right) (100) \]

### Step 5: Simplify the equation
Calculating further:
\[ \Delta U = 2 \cdot \frac{3R}{2} \cdot 100 \]
The 2s cancel out:
\[ \Delta U = 3R \cdot 100 = 300R \]

### Step 6: Conclude the result
Since \( \Delta Q = \Delta U \), we have:
\[ \Delta Q = 300R \]

Thus, the amount of heat needed to raise the temperature of the gas is:
\[ \Delta Q = 300R \]

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

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