Solve Isothermal Expansion - Pressure, Volume & Work Explained

In summary, to find the external work performed during the isothermal expansion of air with an initial pressure of 40 psig and initial volume of 8 cu. ft. to a final pressure of 10 psig, you can use the formula PV=nRT. With constant temperature, P_iV_i = P_fV_f, so you can solve for the final volume. Then, the work performed is given by the integral of PdV which can be simplified to nRT times the natural logarithm of the final volume divided by the initial volume. No conversions are necessary, just the pressure ratio P_f/P_i. However, if working in MKS units, conversions may be needed before and after finding the pressure ratio.
  • #1
shawn100
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If air has a pressure of 40 psig and a volume of 8 cu. ft. expands isothermally to a pressure of 10 psig, find the external work performed during the expansion. How do I do this, do I first have to change 40 psig to psia, and how do I do that? This question has me lost! Any help appreciated, even a formula for me to understand it would help. thank you
 
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  • #2
shawn100 said:
If air has a pressure of 40 psig and a volume of 8 cu. ft. expands isothermally to a pressure of 10 psig, find the external work performed during the expansion. How do I do this, do I first have to change 40 psig to psia, and how do I do that? This question has me lost! Any help appreciated, even a formula for me to understand it would help. thank you
Use PV=nRT.

If T is constant then [itex]P_iV_i = P_fV_f[/itex]. So you can work out what the final volume is.

The work is:

[tex]W = \int_{P_i}^{P_f} PdV = \int_{V_i}^{V_f} \frac{nRT}{V}dV [/tex]

You have to work out that integral (hint: [itex]\frac{d}{dV}ln V = 1/V[/itex]) and plug in the initial and final volumes.

You don't have to do any conversions. You just need the pressure ratio P_f/P_i.

AM

[edit: this last comment is not quite correct. You do have to work out nRT = P_iV_i which means you have to do a conversion. PSIA is absolute pressure in pounds/in^2, which means you have to include atmospheric pressure. PSIG is gauge pressure, which is 1 atm less than actual. It is easier to work in MKS. I would convert to MKS and then convert back.]

AM
 
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  • #3


To solve this problem, we first need to understand the concept of isothermal expansion. This is a process in which a gas expands at a constant temperature, meaning that there is no change in the internal energy of the gas. In this case, the air is expanding from an initial pressure of 40 psig to a final pressure of 10 psig while maintaining a constant temperature.

To find the external work performed during this isothermal expansion, we can use the formula W = -PΔV, where W is the work done, P is the pressure, and ΔV is the change in volume. However, before we can use this formula, we need to convert the pressure from psig (pounds per square inch gauge) to psia (pounds per square inch absolute).

To convert from psig to psia, we need to add the atmospheric pressure to the given pressure. Atmospheric pressure is the pressure exerted by the Earth's atmosphere, which is approximately 14.7 psia at sea level. So, for our problem, we would add 14.7 psia to the initial pressure of 40 psig, giving us a total pressure of 54.7 psia. We would then do the same for the final pressure, adding 14.7 psia to 10 psig to get a final pressure of 24.7 psia.

Now, we can plug these values into our formula: W = -PΔV = -(54.7 psia - 24.7 psia)(8 cu. ft.) = -30 psia * 8 cu. ft. = -240 ft-lbf

This means that the external work performed during the isothermal expansion is -240 ft-lbf (foot-pounds). The negative sign indicates that the work is done on the gas, rather than by the gas. In other words, the gas is being compressed by an external force, rather than expanding on its own.

I hope this explanation helps you understand how to solve this problem. If you have any further questions, please don't hesitate to ask. Good luck!
 

1. What is isothermal expansion?

Isothermal expansion is a process in thermodynamics where the volume of a gas increases while the temperature remains constant.

2. How is pressure related to isothermal expansion?

In isothermal expansion, the pressure of the gas decreases as the volume increases. This is because the temperature remains constant, so according to the ideal gas law, PV = nRT, if volume increases, pressure must decrease to maintain a constant temperature.

3. What is the equation for isothermal expansion?

The equation for isothermal expansion is PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

4. How does work relate to isothermal expansion?

Work is done during isothermal expansion because the gas expands against the external pressure. The work done is equal to the product of the external pressure and the change in volume.

5. What are some real-life examples of isothermal expansion?

Some real-life examples of isothermal expansion include the operation of a steam engine, the expansion of a balloon filled with helium, and the expansion of a gas in a tank with a movable piston.

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