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Question: The amount of heat energy required to raise the temperature of \(1g\) of Helium at NTP, from \({T_1}...

The amount of heat energy required to raise the temperature of 1g1g of Helium at NTP, from T1K{T_1}K to T2K{T_2}K is ?
A) 32NaKB(T2−T1)\dfrac{3}{2}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)
B) 34NaKB(T2−T1)\dfrac{3}{4}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)
C) 34NaKB(T2T1)\dfrac{3}{4}{N_a}{K_B}\left( {\dfrac{{{T_2}}}{{{T_1}}}} \right)
D) 38NaKB(T2−T1)\dfrac{3}{8}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)

Explanation

Solution

For solving this question the mass of the HeHe is given in the equation and we know that molar mass of the HeHe is 4g/mol4g/mol by this we get the number of moles of HeHe . And specific heat at constant volume is 32R\dfrac{3}{2}R and then change in temperature that is ΔT=T2−T1\Delta T = {T_2} - {T_1} , by putting all these values in Δθ=nCvΔT\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=1gmHe = 1gm
Molar mass of He=4g/molHe = 4g/mol
No of moles of the heat=mM = \dfrac{m}{M}
=14mol= \dfrac{1}{4}mol
And given rising temperature will be ΔT=T2−T1\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 Cv{C_v} for Helium is
And Cv=32R{C_v} = \dfrac{3}{2}R because it is monoatomic gas.
Now, ΔQ=nCvΔT\Delta Q = n{C_v}\Delta T
Now, putting the values.
=14×32R×(T2−T1)= \dfrac{1}{4} \times \dfrac{3}{2}R \times \left( {{T_2} - {T_1}} \right)
=38R(T2−T1)= \dfrac{3}{8}R\left( {{T_2} - {T_1}} \right)
Now, substituting the value.
=38NaKB(T2−T1)= \dfrac{3}{8}{N_a}{K_B}\left( {{T_2} - {T_1}} \right)
Where R=R = Boltzmann constant KB{K_B} ×\times Avogadro number Na{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=KB×NaR = {{\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.