Near the surface of Venus, the rms speed of carbon diox ide molecules (`CO_2`) is 650 m/s. What is the temperature (in kelvins) of the atmosphere at that point?

Options

A. 250 K
B. 920 K
C. 750 K
D. 1800 K

Answer

To find the temperature of the atmosphere on Venus given the root mean square (rms) speed of carbon dioxide molecules (CO₂), we can use the formula for the rms speed:

\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

Where:
- \( V_{rms} \) is the root mean square speed,
- \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \)),
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 1: Square the rms speed
First, we square the rms speed to eliminate the square root:

\[ V_{rms}^2 = \frac{3RT}{M} \]

Given \( V_{rms} = 650 \, \text{m/s} \):

\[ 650^2 = \frac{3RT}{M} \]

Calculating \( 650^2 \):

\[ 650^2 = 422500 \, \text{m}^2/\text{s}^2 \]

### Step 2: Calculate the molar mass of CO₂
The molar mass of CO₂ can be calculated as follows:

- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol, and there are 2 oxygen atoms in CO₂.

Thus, the molar mass \( M \) is:

\[ M = 12 + 2 \times 16 = 12 + 32 = 44 \, \text{g/mol} \]

To convert this to kg/mol:

\[ M = 44 \times 10^{-3} \, \text{kg/mol} \]

### Step 3: Rearrange the equation to solve for temperature \( T \)
Rearranging the equation to solve for \( T \):

\[ T = \frac{M \cdot V_{rms}^2}{3R} \]

### Step 4: Substitute the known values into the equation
Substituting \( M = 44 \times 10^{-3} \, \text{kg/mol} \), \( V_{rms}^2 = 422500 \, \text{m}^2/\text{s}^2 \), and \( R = 8.314 \, \text{J/(mol K)} \):

\[ T = \frac{(44 \times 10^{-3}) \cdot 422500}{3 \cdot 8.314} \]

### Step 5: Calculate the temperature
Now we calculate the value:

1. Calculate the numerator:

\[ 44 \times 10^{-3} \cdot 422500 = 18695 \]

2. Calculate the denominator:

\[ 3 \cdot 8.314 = 24.942 \]

3. Now divide the numerator by the denominator:

\[ T = \frac{18695}{24.942} \approx 749.32 \, \text{K} \]

### Step 6: Round the temperature
Rounding the temperature gives us:

\[ T \approx 750 \, \text{K} \]

### Final Answer
The temperature of the atmosphere near the surface of Venus is approximately **750 K**.

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