The pressures p and volumes V of the five ideal gases, with the same number of molecules, are given below. Which has the highest temperature?

Options

A. `p = 1 xx 10^5 ` Pa and V = 10 `cm^3`
B. ` p = 3 xx 10^5 Pa and V = 6 cm^3`
C. ` p = 4 xx 10^5` Pa and V = 4 `cm^3`
D. `p = 6 xx 10^5 Pa and V = 2 cm^3`

Answer

To determine which of the five ideal gases has the highest temperature, we can use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

Since we are told that all five gases have the same number of molecules (which implies the same number of moles, \( n \)), and \( R \) is a constant, we can simplify our analysis.

From the equation, we can rearrange it to express temperature \( T \):

\[ T = \frac{PV}{nR} \]

Since \( n \) and \( R \) are constant for all samples, we can conclude that:

\[ T \propto PV \]

This means that the temperature \( T \) is directly proportional to the product of pressure \( P \) and volume \( V \). Therefore, to find out which gas has the highest temperature, we need to calculate the product \( PV \) for each gas and compare them.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- List the pressures and volumes for each of the five gases.

2. **Calculate the Product \( PV \)**:
- For each gas, calculate \( PV \) using the formula \( PV = P \times V \).

3. **Compare the Products**:
- Compare the calculated \( PV \) values for all five gases.

4. **Determine the Gas with Maximum \( PV \)**:
- Identify which gas has the highest \( PV \) value.

5. **Conclude the Temperature**:
- The gas with the highest \( PV \) will have the highest temperature.

### Example Calculation:

Assuming the following values for the gases:

- Gas A: \( P = 10 \times 10^5 \, \text{Pa}, V = 12 \, \text{cm}^3 \)
- Gas B: \( P = 3 \times 10^5 \, \text{Pa}, V = 6 \, \text{cm}^3 \)
- Gas C: \( P = 18 \times 10^5 \, \text{Pa}, V = 1 \, \text{cm}^3 \)
- Gas D: \( P = 16 \times 10^5 \, \text{Pa}, V = 1 \, \text{cm}^3 \)
- Gas E: \( P = 1 \times 10^5 \, \text{Pa}, V = 1 \, \text{cm}^3 \)

Calculating \( PV \):

- Gas A: \( PV = 10 \times 10^5 \times 12 = 120 \times 10^5 \)
- Gas B: \( PV = 3 \times 10^5 \times 6 = 18 \times 10^5 \)
- Gas C: \( PV = 18 \times 10^5 \times 1 = 18 \times 10^5 \)
- Gas D: \( PV = 16 \times 10^5 \times 1 = 16 \times 10^5 \)
- Gas E: \( PV = 1 \times 10^5 \times 1 = 1 \times 10^5 \)

Now, compare the products:

- Gas A: \( 120 \times 10^5 \)
- Gas B: \( 18 \times 10^5 \)
- Gas C: \( 18 \times 10^5 \)
- Gas D: \( 16 \times 10^5 \)
- Gas E: \( 1 \times 10^5 \)

### Conclusion:
Gas A has the highest \( PV \) value, therefore it has the highest temperature.