Answer:
∆T = Mv^2Y/2Cp
Explanation:
Formula for Kinetic energy of the vessel = 1/2mv^2
Increase in internal energy Δu = nCVΔT
where n is the number of moles of the gas in vessel.
When the vessel is to stop suddenly, its kinetic energy will be used to increase the temperature of the gas
We say
1/2mv^2 = ∆u
1/2mv^2 = nCv∆T
Since n = m/M
1/2mv^2 = mCv∆T/M
Making ∆T subject of the formula we have
∆T = Mv^2/2Cv
Multiple the RHS by Cp/Cp
∆T = Mv^2/2Cv *Cp/Cp
Since Y = Cp/CV
∆T = Mv^2Y/2Cp k
Since CV = R/Y - 1
We could also have
∆T = Mv^2(Y - 1)/2R k
Answer:
ΔT = (Y-1/2R) Mv² Kelvin, whereΔT = Change in temperature
Explanation:
Kinetic energy of vessel = 1/2mv², Change in internal energy(ΔU) = nCvΔT
n = number of moles of gas in vessel
When the vessel is sudeenly stopped, its kinetic energy causes a rise in the temperature of the gas
∴ 1/2mv² = ΔU
ΔU = nCVΔT
CP -CV = R, from CP/CV = Y, CP = CV Y
CP - CV = R
CVY - CV = R
CV(Y-1) = R
CV = R/Y-1
Recall that 1/2mv² = ΔU = nCVΔT
nCVΔT = 1/2mv²
n( R/Y-1)ΔT = 1/2mv² ⇔ substituting the value of CV above
from n= mass(m)/Molar mass(M), ⇔ m = nM
we have ,
n (R/Y-1)ΔT = 1/2nMv²
canceling out n
∴ ΔT = (Y-1/2R) Mv² Kelvin , the change in temperature
