Heat is added isothermally to 2.5 mol of a monatomic ideal gas. The temperature of the gas is 430 K. How much heat must be added to make the volume of the gas double?

Options

A. 5500 J
B. 3100 J
C. 6700 J
D. 2700 J

Answer

To solve the problem of how much heat must be added to a monatomic ideal gas to double its volume isothermally, we can follow these steps:

### Step 1: Understand the process
Since the process is isothermal, the temperature remains constant. For an ideal gas undergoing an isothermal process, the change in internal energy (\( \Delta U \)) is zero because internal energy is a function of temperature.

### Step 2: Apply the First Law of Thermodynamics
The First Law of Thermodynamics states:
\[ \Delta Q = \Delta U + W \]
For an isothermal process, since \( \Delta U = 0 \):
\[ \Delta Q = W \]
This means that the heat added to the system is equal to the work done by the system.

### Step 3: Use the formula for work done in an isothermal process
The work done (\( W \)) by an ideal gas during an isothermal expansion can be calculated using the formula:
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
where:
- \( n \) = number of moles of the gas
- \( R \) = ideal gas constant (8.314 J/(mol·K))
- \( T \) = absolute temperature in Kelvin
- \( V_f \) = final volume
- \( V_i \) = initial volume

### Step 4: Substitute the values
Given:
- \( n = 2.5 \) mol
- \( T = 430 \) K
- \( V_f = 2V_i \) (the final volume is double the initial volume, so \( \frac{V_f}{V_i} = 2 \))

Now substituting these values into the work done formula:
\[ W = 2.5 \times 8.314 \times 430 \times \ln(2) \]

### Step 5: Calculate the values
First, calculate \( \ln(2) \):
\[ \ln(2) \approx 0.693 \]

Now substitute this into the equation:
\[ W = 2.5 \times 8.314 \times 430 \times 0.693 \]

Calculating this step-by-step:
1. Calculate \( 2.5 \times 8.314 \):
\[ 2.5 \times 8.314 \approx 20.785 \]
2. Calculate \( 20.785 \times 430 \):
\[ 20.785 \times 430 \approx 8938.55 \]
3. Finally, calculate \( 8938.55 \times 0.693 \):
\[ 8938.55 \times 0.693 \approx 6205.47 \text{ J} \]

### Step 6: Conclusion
Thus, the amount of heat that must be added to the gas to double its volume is approximately:
\[ \Delta Q \approx 6205.47 \text{ J} \]