At what temperature , will the rms speed of oxygen molecules be sufficient for escaping from the earth ? Take `m = 2.76 xx 10^(-26) kg, k = 1.38 xx 10^(-23) J//K and v_(e) = 11.2 km//s`.

Options

A. `5.016xx10^(4) K`
B. `8.326xx10^(4)K`
C. `2.2508xx10^(4) K`
D. `1.254xx106(4) K`

Answer

To find the temperature at which the root mean square (rms) speed of oxygen molecules is sufficient for escaping from the Earth, we can use the formula for the rms speed of gas molecules:

\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]

where:
- \( v_{rms} \) is the root mean square speed,
- \( k \) is the Boltzmann constant,
- \( T \) is the temperature in Kelvin,
- \( m \) is the mass of a single molecule.

We know that the escape velocity \( v_e \) from the Earth is given as \( 11.2 \, \text{km/s} \), which we need to convert to meters per second:

\[ v_e = 11.2 \, \text{km/s} = 11200 \, \text{m/s} \]

Setting the rms speed equal to the escape velocity, we have:

\[ \sqrt{\frac{3kT}{m}} = v_e \]

Squaring both sides gives:

\[ \frac{3kT}{m} = v_e^2 \]

Now, we can solve for \( T \):

\[ T = \frac{m v_e^2}{3k} \]

Next, we will substitute the values into the equation. Given:
- \( m = 2.76 \times 10^{-26} \, \text{kg} \)
- \( k = 1.38 \times 10^{-23} \, \text{J/K} \)
- \( v_e = 11200 \, \text{m/s} \)

Now, we can calculate \( v_e^2 \):

\[ v_e^2 = (11200)^2 = 125440000 \, \text{m}^2/\text{s}^2 \]

Now substituting the values into the equation for \( T \):

\[ T = \frac{(2.76 \times 10^{-26} \, \text{kg}) \times (125440000 \, \text{m}^2/\text{s}^2)}{3 \times (1.38 \times 10^{-23} \, \text{J/K})} \]

Calculating the numerator:

\[ (2.76 \times 10^{-26}) \times (125440000) = 3.465984 \times 10^{-18} \, \text{kg m}^2/\text{s}^2 \]

Now calculating the denominator:

\[ 3 \times (1.38 \times 10^{-23}) = 4.14 \times 10^{-23} \, \text{J/K} \]

Now substituting these values into the equation for \( T \):

\[ T = \frac{3.465984 \times 10^{-18}}{4.14 \times 10^{-23}} \approx 8.36 \times 10^4 \, \text{K} \]

Thus, the temperature at which the rms speed of oxygen molecules is sufficient for escaping from the Earth is approximately:

\[ T \approx 8.36 \times 10^4 \, \text{K} \]