# Thermodynamics

#### Bigron1

I've got a question asking work out thermal transfer and work done using Charles Law when final temp is 250k where P = ( v - 4 )^2. 5kg gas expands from 300 to 400mm^3. Cp = 1005j/kg K.

Its the P = ( v - 4)^2 bit that is throwing me.

Thanks for any help

#### benit13

Your question is a little odd. Do you have the original question? Can you please scan it and upload it or link it?

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#### Bigron1

5kg of a gas expands from 0.3to 0.4 m3 according to Charles law evaluate thermal energy transfer & work done for the gas. Cp =1005 J/kg K. final gas
temp is 250K where P = ( v-4 )2. That’s squared not sure how to do a square symbol on here.

#### benit13

Sorry for the late reply... I've been quite busy at work over the past few days. My thermodynamics is also a bit rusty because I haven't done any for a while, so feel free to point out any issues.

I think there's also some issues with the question...

You state that the expansion occurs due to Charle's law, which indicates that the pressure is held constant during the gas expansion so that:

$$\displaystyle \frac{V_1}{T_1} = \frac{V_2}{T_2}$$

However, you also state that there is a relationship between pressure and volume of the form

$$\displaystyle P = (V - 4)^2$$

This law is very odd because it seems to suggest that the pressure is negative for the expansion states when $$\displaystyle V_1$$ or $$\displaystyle V_2$$ is substituted into the law.

Is what you typed exactly what the question states? If so, it is an awful question.

If we continue under the assumption that Charles' law is valid, then:

mass = $$\displaystyle m = 5$$ kg
heat capacity = $$\displaystyle 1005$$ J/kg/K

Iniitial state:
$$\displaystyle V_1 = 0.3 m^3$$
$$\displaystyle T_1 = ?$$
$$\displaystyle P_1 = ?$$

Final state:
$$\displaystyle V_2 = 0.4 m^3$$
$$\displaystyle T_2 = 250$$ K
$$\displaystyle P_2 = ?$$

Under isobaric expansion, we can find $$\displaystyle T_1$$:

$$\displaystyle T_1 = \frac{V_1 T_2}{V_2} = \frac{0.3 \times 250}{0.4} = 187.5$$ K.

The thermal energy transfer to the gas can be calculated using the equation for sensible heat transfer:

$$\displaystyle \Delta Q = m c_p (T_2 - T_1)= 5 \times 1005 (250 - 187.5) = 314062.5$$ J

Finally, the work done by the gas is equal to the area underneath the pressure-volume curve for the expansion. If pressure is constant

$$\displaystyle W = \int - P dV = - P \Delta V$$

But we don't know the pressure. In order to go further, we have to make assumptions about the kind of gas we have. A heat capacity of 1005 J/kg/K suggests that we have dry air, which has a molar mass of 0.029 kg/mol. If we assume that air behaves like an ideal gas, we have:

$$\displaystyle P = \frac{mRT_2}{V_2 M} = \frac{5 \times 8.314 \times 250}{0.4 \times 0.029} = 895905$$ Pa

This means that the work done is

$$\displaystyle W = - 895905 (0.4 - 0.3) = - 89590.5$$ J

The negative sign indicates that the work is done by the gas on the surroundings.

Note: If we actually have a two-part question and the expansion is not isobaric, we can instead use stated pressure-relationship and we have

$$\displaystyle W = \int -P dV$$
$$\displaystyle = -\int_{0.3}^{0.4} (V-4)^2 dV$$
$$\displaystyle = -\int_{0.3}^{0.4} (V^2 - 8V + 16) dV$$
$$\displaystyle = -\left[ \frac{V^3}{3} - 4V^2 + 16V \right]_{0.3}^{0.4}$$
$$\displaystyle = -\left(0.0213 - 0.64 + 6.4 - 0.009 + 0.36 - 4.8\right)$$
$$\displaystyle = -1.3323$$ J

This result is really off to me. It could be that the question you have just wants to test your ability to perform integration, but it's inconsistent with the rest of the question and gives a result that is not realistic at all.

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#### Bigron1

Thanks mate
Yes the question is very vague in describing what kind of process it goes through. I don't thinks its a genuine question trying to get an actual real world value just testing to see if you can use what you've learnt to get an answer.
To answer your other question about a second part it also asks to evaluate the same for Boyles law. How do you only use one to get the answer for work done and thermal transfer? I thought you needed to combine them?

Blimey you answer posts late at night or early morning depending which way you look at it. zzzz

Thanks for the help

#### benit13

To answer your other question about a second part it also asks to evaluate the same for Boyles law. How do you only use one to get the answer for work done and thermal transfer? I thought you needed to combine them?
Boyle's law states that there is a relationship between pressure and volume when the temperature is held constant.

$$\displaystyle P \propto \frac{1}{V}$$

Therefore

$$\displaystyle P_1 V_1 = P_2 V_2$$

Our previous calculation for pressure uses $$\displaystyle T_2$$ and $$\displaystyle V_2$$, so if we say that the pressure there is $$\displaystyle P_2$$, we can calculate $$\displaystyle P_1$$

$$\displaystyle P_1 = \frac{P_2 V_2}{V_1} = \frac{895905 \times 0.4}{0.3} = 1194540$$Pa.

The thermal energy exchange is zero because the temperature doesn't change. We used the value T = 250 K.

The work done can be calculated using integration:

$$\displaystyle W = \int -P dV$$
$$\displaystyle = -\frac{mRT}{M} \int_{0.3}^{0.4} \frac{1}{V} dV$$
$$\displaystyle = -\frac{mRT}{M} \left[ \ln V \right]_{0.3}^{0.4}$$
$$\displaystyle = -\frac{mRT}{M} \left( \ln 0.4 - \ln 0.3 \right)$$
$$\displaystyle = -\frac{mRT}{M} \frac{\ln 0.4}{\ln 0.3}$$
$$\displaystyle = -\frac{5 \times 8.314 \times 250}{0.029} \frac{\ln 0.4}{\ln 0.3}$$
$$\displaystyle = - 272733.6$$ J

Blimey you answer posts late at night or early morning depending which way you look at it. zzzz

Thanks for the help
I am based on the UK, so the difference in time-zone probably makes my post appear as if they are at unusual hours. However, I can assure you I'm not up in the middle of the night solving physics problems! #### Bigron1

Sorry for the slow reply Benit busy weekend.
Hopefully final question for you. If I am supposed to integrate the P for both questions wouldn't I get the same answer for both questions? But then if I was supposed to do that then I would breaking the gas law what I am supposed to be using to work out the answer. Very confusing.
Is that kind of what you are getting at?

Cheers Bud

#### benit13

Sorry for the slow reply Benit busy weekend.
Hopefully final question for you. If I am supposed to integrate the P for both questions wouldn't I get the same answer for both questions? But then if I was supposed to do that then I would breaking the gas law what I am supposed to be using to work out the answer. Very confusing.
Is that kind of what you are getting at?

Cheers Bud
You won't get the same answer because one expansion is isobaric and the other is isothermal. The initial conditions required to achieve the final conditions under each case are different.

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#### Bigron1

My tutor said I used the in correct method to calculate work done in both parts.
I've looked at it again and
For Charles Law question is work done simply W=P∆V?
So pressure = 0.4-4^2 = 12.96 -- W=P∆V = 12.96x0.1= 1.296J
The second part of the second in Boyles law assuming its isothermal is work done = -mRT In(v2/v1) so -5*8.31447*250 = 73.229J

This might be a very simplistic way to look at it but I'm only supposed to be doing a basic intro into therm0dynamics course.

Any thoughts?

Thanks

#### benit13

My tutor said I used the in correct method to calculate work done in both parts.

I've looked at it again and
For Charles Law question is work done simply W=P∆V?
Yes. That law is valid for cases when the pressure is held constant.

https://en.wikipedia.org/wiki/Charles's_law

So pressure = 0.4-4^2 = 12.96 -- W=P∆V = 12.96x0.1= 1.296J
I never understood the quadratic formula

$$\displaystyle P = (V-4)^2$$

and how it fits with the question, so if your tutor explains it, then that's fine. However, if that formula describes the change in pressure-volume curve of the state transition, then the expansion is not conforming to Charles' law because pressure is not constant... it's varying with volume.

The general formula for work done is

$$\displaystyle dW = - P(V) dV$$

If pressure is constant with the volume change, then this reduces to

$$\displaystyle \Delta W = -P \Delta V$$

Otherwise you have to solve

$$\displaystyle \Delta W = \int -P(V) dV$$

So I would confirm with your tutor whether the pressure is constant or not.

The second part of the second in Boyles law assuming its isothermal is work done = -mRT In(v2/v1) so -5*8.31447*250 = 73.229J
The only difference between this formula and mine is that I've included the molar mass of the gas (0.029). The ideal gas law using the molar gas constant is written as follows

$$\displaystyle PV = nRT$$

where n is the number of moles of gas, not the mass. The number of moles of gas can be estimated using the ratio (m/M) where m is the mass of the gas (kg) and M is the molar mass of the gas (kg/mol). I think the tutor's formula neglects the molar mass on the denominator.

This might be a very simplistic way to look at it but I'm only supposed to be doing a basic intro into therm0dynamics course.

Any thoughts?

Thanks
It's fine. Disagreements happen from time to time, but they are usually resolvable. If it gets out of hand, it's usually not worth pressing the issue too hard and it's better instead, imho, to revisit it later.

Also... if this question is from a book, can you post the reference here for it?

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