Correct Answer - Option 4 : 25 R
CONCEPT:
Internal energy
- Internal energy is the total energy of the gas due to the random motion of the particles along with the potential energies of the molecules due to their orientation.
- The internal energy of the gas increases as the temperature of the gas increases.
- The increase in internal energy with temperature is given as,
\(\Rightarrow Δ U=nC_{v}Δ T\)
Where ΔU = increase in internal energy, n = number of moles, Cv = specific heat at constant volume and Δt = increase in temperature
The first law of thermodynamics
- This law states that the total heat given or taken out of the system will always be equal to the work done and change in the internal energy of the system.
⇒ ΔQ = ΔW + ΔU
CALCULATION:
Given n = 1, ΔT = 10° C and V1 = V2 = V (Constant volume process)
- We know that for the thermodynamic process the work done is given as,
\(\Rightarrow Δ W=PΔ V\) -----(1)
Where P = pressure and ΔV = change in volume
- For a constant volume process,
\(\Rightarrow Δ V=0\) -----(2)
By equation 1 and equation 2,
\(\Rightarrow Δ W=0 J\) -----(3)
- By the first law of thermodynamics,
\(\Rightarrow Δ Q=Δ W+Δ U\) -----(4)
Where ΔQ = heat added to the gas
By equation 3 and equation 4,
\(\Rightarrow Δ Q=0+Δ U\)
\(\Rightarrow Δ Q=nC_{v}\Delta T\) -----(5)
- We know that for diatomic gas,
\(\Rightarrow C_{v}=\frac{5}{2}R\) -----(6)
By equation 5 and equation 6,
\(\Rightarrow Δ Q=1\times\frac{5}{2}R\times10\)
\(\Rightarrow Δ Q=25R\)
- Hence, option 4 is correct.
