Suppose you have a mixture of hydrogen and methane (CH4) gases that are at equilibrium. What is the ratio of the average speed of the methane (CH4) relative to the hydrogen ? Express the answer as a fraction (e.g. 0.621).
I know that im suppose to use the e=1/2mv2 but i don't know what goes where
To find the ratio of the average speed of methane (CH4) relative to hydrogen in a mixture, you need to compare their average speeds using the equation for kinetic energy:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the particle, and v is the velocity of the particle.
In this case, let's assume both hydrogen and methane molecules have the same mass, denoted as m. Therefore, their kinetic energies can be expressed as:
KE(H2) = (1/2) * m * v(H2)^2
KE(CH4) = (1/2) * m * v(CH4)^2
The kinetic energy of a particle is directly proportional to the square of its velocity. Therefore, we can write:
KE(H2) / KE(CH4) = (v(H2)^2) / (v(CH4)^2)
Since the masses (m) cancel out, we can simplify the equation as:
v(H2)^2 / v(CH4)^2 = KE(H2) / KE(CH4)
However, we know that both hydrogen and methane gases are at equilibrium, which means they are at the same temperature. According to the kinetic theory of gases, at the same temperature, the kinetic energy of gas molecules is directly proportional to their absolute temperature (T).
Therefore, we can write:
KE(H2) / KE(CH4) = T(H2) / T(CH4)
The ratio of the average speeds can now be expressed as:
(v(H2) / v(CH4))^2 = T(H2) / T(CH4)
Now, we need to consider the molar masses (M). The molar mass of hydrogen (H2) is approximately 2 g/mol, and the molar mass of methane (CH4) is approximately 16 g/mol.
The ratio of the average speed can now be written as:
(v(H2) / v(CH4))^2 = sqrt(M(CH4) / M(H2))
Substituting the values:
(v(H2) / v(CH4))^2 = sqrt(16 g/mol / 2 g/mol) = sqrt(8) = √8
Taking the square root:
(v(H2) / v(CH4)) = √8
To express the answer as a fraction, you can rationalize the denominator:
(v(H2) / v(CH4)) = (√8 * √8) / (√8) = 8 / √8 = 8 / 2√2 = 4√2 / 2√2 = 4/2 = 2
Therefore, the ratio of the average speed of methane (CH4) relative to hydrogen is 2.