Question
Download Solution PDFA thermally insulated vessel contains an ideal gas of molecular mass M anda ratio of specific heat 1.4. The vessel is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :
(R = universal gas constant)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCONCEPT:
- Specific heat is defined as the heat required to raise the temperature of the unit mass of a given substance by a given amount and it is written as;
E = nCΔT ( In short, this is the energy gain in the system)
CALCULATION:
Given: γ = 1.4
Let us take "n" moles of the gas.
The energy loss is equal to the energy gain in the system and it is written as;
Eloss = Egain -----(1)
The energy loss is written as Eloss = \(\frac{1}{2}\)nMv2
and the energy gain is written as Egain = nCvΔT
From, the equation we have;
Eloss = Egain
⇒ \(\frac{1}{2}\)nMv2 = nCvΔT
⇒ \(\frac{1}{2}\)Mv2 = CvΔT ----(2)
γ = 1.4 = 7/5 (given), from this we see that it is a diatomic gas, and specific heat capacity at constant volume is written as;
Cv = 5R/2
Here equation (2) becomes,
\(\frac{1}{2}\)Mv2 = \(\frac{5R}{2}\)ΔT
⇒ ΔT = \(\frac{Mv^2}{5R}\)
Hence, option 2) is the correct answer.
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