Question

The amount of heat energy required to raise the temperature of $1g$ of Helium at NTP, from ${T_1}K$ to ${T_2}K$ is ?
A) $\dfrac{3}{2}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)$
B) $\dfrac{3}{4}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)$
C) $\dfrac{3}{4}{N_a}{K_B}\left( {\dfrac{{{T_2}}}{{{T_1}}}} \right)$
D) $\dfrac{3}{8}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)$

Answer 1

Hint: For solving this question the mass of the $He$ is given in the equation and we know that molar mass of the $He$ is $4g/mol$ by this we get the number of moles of $He$ . And specific heat at constant volume is $\dfrac{3}{2}R$ and then change in temperature that is $\Delta T = {T_2} - {T_1}$ , by putting all these values in $\Delta \theta = n{C_v}\Delta T$ we get the heat required.

Complete step by step answer:
We know that,
Heat is defined as it is form of energy which is used to transfer Kinetic energy from medium or object to another this is known as Heat.
Mass of $He = 1gm$
Molar mass of $He = 4g/mol$
No of moles of the heat$ = \dfrac{m}{M}$
$ = \dfrac{1}{4}mol$
And given rising temperature will be $\Delta T = {T_2} - {T_1}$
Temperature is defined as it is a physical quantity that is used to express hot and cold means whether the temperature is hot or cold. This is known as Temperature.
So, the value of ${C_v}$ for Helium is
And ${C_v} = \dfrac{3}{2}R$ because it is monoatomic gas.
Now, $\Delta Q = n{C_v}\Delta T$
Now, putting the values.
$ = \dfrac{1}{4} \times \dfrac{3}{2}R \times \left( {{T_2} - {T_1}} \right)$
$ = \dfrac{3}{8}R\left( {{T_2} - {T_1}} \right)$
Now, substituting the value.
$ = \dfrac{3}{8}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)$
Where $R = $Boltzmann constant ${K_B}$ $ \times $ Avogadro number ${N_a}$
Boltzmann constant is defined as it is a type of physical constant that relates a particle energy with temperature.
Avogadro number is defined as it is an absolute number. The number of units in one mole for any substance is known as Avogadro number.
Which means $R = {{\rm K}_B} \times {N_a}$
Hence, the correct option is D.

Note: It is important to consider 1g of helium instead of 1mole of Helium in this question as mentioned as it can lead to calculation error. Also the equations in kinetic theory of gases are only valid for ideal gases.