In which case does the free path not affected?
A.${H_2}$ gas at ${100^o}C$ and $1atm$ is transferred into a vessel at ${50^o}C$ and $0.5atm$
B.${O_2}$ gas at $200K$ and $2atm$ is transferred into a vessel at $400K$ and $1atm$
C.${O_2}$ gas is replaced by ${H_2}$ gas keeping the other variables constant
D.Medium is made more viscous

Hint: In the given question firstly we have to define the characteristics that affect and govern the changes in the mean free path. The mean free path is basically the change in the absolute path of the molecule of the gas. Now for it to be constant the $\dfrac{P}{T}$ need to be constant as well.

Complete answer:
-First thing to keep in mind is that we just have to focus on the characteristics leading to the decision of the mean free path which are mainly the pressure and the temperature.
-For any case to get the mean free path unaffected it needs to get the ratio of the pressure and the temperature constant and unaffected. So we just have to analyze the cases chance by chance.
-The conditions here is : ${H_2}$ gas at ${100^o}C$ and $1atm$ is transferred into a vessel at ${50^o}C$ and $0.5atm$
-In this we get their the ratio for the first case as :
$\dfrac{1}{{100}}$
-And for the second case it is :
$\dfrac{{0.5}}{{50}} = \dfrac{1}{{100}}$
-Now this means the mean free path will not get affected on the given changes for the thermodynamic process.
So by this observation we can say that the correct answer would be option A, ${H_2}$ gas at ${100^o}C$ and $1atm$ is transferred into a vessel at ${50^o}C$ and $0.5atm$.

Note:
In physics, the mean free path is the average distance travelled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modifies its direction or energy or other particle properties.