Question

I. The kinetic energy of argon at ${\mathbf{100K}}$ is approximately equal to the kinetic energy an equal number of neon atoms at ${\mathbf{50K}}$ .
II. A particle with two times the mass of a smaller particle has two times the kinetic energy.
(A) Statement I is true, statement II is true and is a correct explanation of the phenomena described in I.
(B) Statement I is true, statement II is false.
(C) Statement I is false, statement II is true.
(D) Statement I is false, statement II is false.

Answer 1

The correct answer to the above question can be given by understanding the kinetic theory of gases. Also, we need to know how the kinetic energy of two gases is dependent on the temperature, pressure and the other factors which are included in it.

Complete step by step answer:
The kinetic theory of gases is a model which describes the thermodynamic behaviour of gases. According to which many other phenomena were established and understood. It assumes that gas contains identical atoms of large numbers, which moves in a constant random motion with a particular average speed. The distance between the two atoms is considered to be larger than the size of the atom itself.
Now the kinetic energy of a gas can be calculated using the below-given formula;
$K = 3/2*\left( {R/{N_A}} \right)*T - \left( 1 \right)$
Where,
$K$ is average kinetic energy $\left( {Joules} \right)$ ,
$R$ is gas constant $\left( {8.314J/mol*K} \right)$ ,
${N_A}$ is Avogadro’s number $(6.022*{10^2}3atoms/mol)$ ,
$T$ is temperature $\left( K \right)$ .
Now we are provided with the temperature of both the gases and we are said that the number of an atom of both the gases is equal.
The kinetic energy of argon:
$T = 100K$ ,
${K_a} = 3/2*\left( {R/{N_A}} \right)*100$
$\Rightarrow$ ${K_a} = 100*\left( {3/2} \right)*\left( {R/{N_A}} \right)$
The kinetic energy of neon:
$T = 50K$
${K_n} = 3/2*\left( {R/{N_A}} \right)*50K$
$\Rightarrow$ $\;{K_n} = 50*\left( {3/2} \right)*\left( {R/{N_A}} \right)$
Finding the ratio of the kinetic energy of both the gases we get:
${K_a}/{K_n} = \left[ {100*\left( {3/2} \right)*\left( {R/{N_A}} \right)} \right]/\left[ {{\;{ }}50*\left( {3/2} \right)*\left( {R/{N_A}} \right)} \right]$
Cancelling all the common terms we get:
${K_a}/{K_n} = 100{N_A}/50{N_A}$
Since the number of atoms of both the gases is equal therefore their ${N_A}$ is also equal and hence,
${K_a}/{K_n} = 100/50$
${K_a}/{K_n} = 2$
Therefore, ${K_a}:{K_n}::2:1$
Therefore, the statement I is true, statement II is true and is a correct explanation of the phenomena described in I.
Correct option is A.

Note:
The kinetic theory of gases is a model which describes the thermodynamic behaviour of gases. According to which many other phenomena were established and understood. The kinetic energy theory was based on the ideal gas law. The real gas does not behave as per ideal gas which was stated by van der Waal.