S2 - S1 > Integral 1-2 dQ/T Irreversible Process. What T ?

[Point Conception]

For an irreversible process: S₂ - S₁ > ∫₁² dQ/T
In the case of heat transfer dQ between object with T₁ to object with T₂, T₁ > T₂
ΔS in Joules/K is known But for evaluating ∫₁² dQ/T = ln dQ/T - ln dQ/T . d/Q is known but what values for T in this case ?
Object 1 & 2 have Temperatures before heat transfer and different temperature after

 

[StarsRuler]

The temperature is variable along the integration, while heat change. Only with the 2 temperatures extremis is not possible use these formula to solve any problem. You need an equation for heat transfer, for example. This is really no thermostatics, it requires thermodynamics, from empirical equations or from statistical mechanics treatments

 

[DrDu]

morrobay, you are right that this formula has quite a restricted range of applicability as T may not be defined during a general irreversible process.

 

[Point Conception]

morrobay, you are right that this formula has quite a restricted range of applicability as T may not be defined during a general irreversible process.

Thanks, would you please check my numerical problem on this in the Homework section;
Introductory Physics. Topic : Entropy for Irreversible Process.

 

[Andrew Mason]

For an irreversible process: S₂ - S₁ > ∫₁² dQ/T

This is incorrect if by dQ you mean the actual heat flow in the process. To calculate the change in entropy, you have to use a reversible path between the beginning and end states

$$\Delta S = \int_{rev path} \frac{dQ_{rev}}{T}$$

AM

 

[Point Conception]

This is incorrect if by dQ you mean the actual heat flow in the process. To calculate the change in entropy, you have to use a reversible path between the beginning and end states

$$\Delta S = \int_{rev path} \frac{dQ_{rev}}{T}$$

AM

Yes it seems that S₂-S₁ > ∫₁²dQ/T
is just qualitative. As Chestermiller showed one way to quantify S₂-S₁
in heat transfer between hotter and cooler object is with.:
Δ S = mC_v(ln T₂/T₁+lnT₂/T₁)
Where the first ln T₂/T₁ is for hotter object losing heat and second
ln T₂/T₁ for cooler object gaining heat.
What is the original integral that this evaluation was derived from ?

 

[Andrew Mason]

Yes it seems that S₂-S₁ > ∫₁²dQ/T
is just qualitative. As Chestermiller showed one way to quantify S₂-S₁
in heat transfer between hotter and cooler object is with.:
Δ S = mC_v(ln T₂/T₁+lnT₂/T₁)
Where the first ln T₂/T₁ is for hotter object losing heat and second
ln T₂/T₁ for cooler object gaining heat.
What is the original integral that this evaluation was derived from ?

It is derived from ΔS = ∫dQ_rev/T:

For the hotter object (body A), a reversible path from T₁ to T₂ would be found by placing it in thermal contact with a body that has an infinitessimally lower temperature and whose temperature keeps decreasing so as to maintain that infinitessimal lower temperature as heat flows out of A, until it reaches T₂. For that path:

$$\Delta S_A = \int_{T_1}^{T_2}\frac{dQ}{T}= \int_{T_1}^{T_2}\frac{mCdT}{T} = mC\ln T_2 - mC\ln T_1 = mC\ln\left(\frac{T_2}{T_1}\right)$$

Similarly the change in entropy of the cooler body, B is:

$$\Delta S_B = mC\ln\left(\frac{T_f}{T_i}\right)$$ where i and f denote initial and final temperatures of Body B.

AM