Easy thermo problem: gas expansion

In summary, the conversation discusses a hypothetical scenario where helium gas expands from an initial volume of 1 liter and an initial pressure of 1 atm to a final volume of 3 liters, with the pressure rising in direct proportion to the volume. The conversation also includes calculations for the work done on the gas, the change in energy content, and the amount of heat added or removed during this process. The solution involves using the ideal gas law and the specific heat capacity of helium to calculate the final energy and heat.
  • #1
EricVT
166
6

Homework Statement



Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

(a) Sketch a graph of pressure versus volume for this process.

(b) Calculate the work done on the gas during this process, assuming that there are no other types of work being done.

(c) Calculate the change in the helium's energy content during this process.

(d) Calculate the amount of heat added to or removed from the helium during this process.

Homework Equations



dQ = W + Q

W = - integral from intial volume to final volume of P(V)dV

The Attempt at a Solution



(a) The graph is just a linear curve with slope 1 from the point (1,1) to (3,3) on the P(atm) vs. V(L) axis.

(b) In SI units we have P(V)= 1.013x10^8 * V (from Pi/Vi and Pf/Vf) so the total work done on the gas is

W= - integral from .001 to .003 of (1.013x10^8)(V)(dv) and this I found to be -405.2 Joules

(c) Since I can't see that any heat was added to or removed from the system is the change in energy just -405.2 Joules?

(d) The heat added to or removed is just zero, isn't it? I don't understand how to know othewise. Unless the net change in energy is zero, then I don't see why heat would be added or removed.

Am I way off here?
 
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  • #2
You've done everything right as far as I can see.
 
  • #3
Actually I did a little more work and the solution I ended up handing in was a bit more complicated.

Since Helium a single atom element, it only has translational kinetic energy contributions, so each Helium atom contributes (3/2)(k)(T) energy to the collective. The entire initial energy would be (N)(3/2)(k)(T)=(3/2)(P)(V) from the ideal gas law. We have both initial pressure and volume, so energy can be calculated here.

Then after the expansion takes place we have the same expression except we plug in our new values of P and V to get final energy. The difference is the change in energy of the system. The work done (calculated above) and the heat added have to equal this change in energy, so heat can be found as such.

Hopefully my second guess/second attempt is right, and it wasn't as easy as I originally thought :/
 
  • #4
You can also try using the specific heat capacity of helium, to calculate the heat entering the system.
 

1. What is the ideal gas law and how does it relate to gas expansion?

The ideal gas law is a mathematical equation that describes the behavior of ideal gases. It states that the pressure (P), volume (V), and absolute temperature (T) of an ideal gas are all directly proportional to each other. This relationship is expressed as P x V = n x R x T, where n is the number of moles of gas and R is the ideal gas constant. Gas expansion occurs when the volume of a gas increases due to a decrease in pressure or an increase in temperature.

2. How does the volume of a gas change during expansion?

The volume of a gas increases during expansion because the gas molecules are moving further apart, resulting in a larger space occupied by the gas. This increase in volume can be caused by a decrease in pressure, an increase in temperature, or a combination of both. The ideal gas law can be used to calculate the change in volume during expansion.

3. What factors affect gas expansion?

The main factors that affect gas expansion are temperature and pressure. An increase in temperature will cause the gas molecules to move faster and increase the volume of the gas. A decrease in pressure will also result in an increase in volume, as the gas molecules have more space to move around. Additionally, the type of gas and its initial conditions (such as volume and pressure) can also affect gas expansion.

4. How is gas expansion measured?

Gas expansion can be measured in terms of the change in volume, pressure, and temperature of the gas. These changes can be measured using various instruments, such as a thermometer to measure temperature, a barometer or manometer to measure pressure, and a graduated cylinder or syringe to measure volume. The ideal gas law can also be used to calculate the change in volume during expansion.

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

Gas expansion occurs in many everyday situations. For example, when a balloon is filled with air and then released, the gas molecules inside the balloon will expand as the pressure decreases, causing the balloon to grow larger. Another example is when a tire is filled with air and then driven on a hot day, the air inside the tire will expand due to the increase in temperature, resulting in an increase in tire pressure. Gas expansion also plays a role in the operation of internal combustion engines, where the expansion of gases from burning fuel causes the engine to produce power.

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