Final answer:
To find the temperature at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, you can use the formulas for root mean square velocity and escape velocity. By equating these two velocities and solving the equation, you can find the temperature. In this case, the temperature is closest to 650 K.
Step-by-step explanation:
To find the temperature at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, we need to equate the two velocities. The root mean square velocity (Urms) can be calculated using the formula Urms = √(3kBT/m), where kB is Boltzmann's constant, T is the temperature, and m is the molar mass of the gas. The escape velocity (Ve) can be calculated using the formula Ve = √(2gR), where g is the acceleration due to gravity and R is the radius of the Earth.
- First, we need to calculate the root mean square velocity of hydrogen using the given values. The molar mass of hydrogen (H₂) is approximately 2.016 g/mol. Plugging these values into the Urms formula, we get Urms = √(3(1.38 × 10⁻²³ J/K)(T)/(0.002016 kg/mol)).
- Next, we need to calculate the escape velocity from the Earth using the given values. Plugging these values into the Ve formula, we get Ve = √(2(10 ms⁻²)(6.4 × 10⁶ m)).
- Now, we equate the two velocities to find the temperature at which they are equal. Setting Urms equal to Ve, we get √(3(1.38 × 10⁻²³ J/K)(T)/(0.002016 kg/mol)) = √(2(10 ms⁻²)(6.4 × 10⁶ m)).
- Simplifying the equation, we find that T ≈ (2(10 ms⁻²)(6.4 × 10⁶ m)(0.002016 kg/mol))/(3(1.38 × 10⁻²³ J/K)).
- Calculating this expression using the given values, we get T ≈ 650 K.
Therefore, the temperature at which the root mean square velocity of hydrogen molecules equals their escape velocity from the Earth is closest to
650 K (option D)