Sample of gas is compressed at a constant temperature

In summary: Since dV/dt is proportional to 1/P^2, it will be greater at the beginning of the 10-minute interval. In summary, when a sample of gas is compressed at a constant temperature, the volume decreases more rapidly at the beginning of 10 minutes than at the end. This is due to the relationship between pressure and volume in the equation PV = C, where pressure increases as volume decreases.
  • #1
Miike012
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Problem: When a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant:

PV = C

Q: A sample of gas is in a container at low pressure and its steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning of 10 minutes or at the end of 10 minutes?

A: dV/dt = -C/P^2(dP/dt)

By looking at my graph... at the end of 10 seconds dV/dt is approaching smaller and smaller values... thus I picked "end of 10 seconds" ... but the answer is beginning of 10 seconds... what did I do wrong?
 

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  • #2
Hmm, the question says "steadily compressed" which I take to mean that dP/dt is a constant function. But... (removed).

Aargh, sorry, it's late, ignore this.
 
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  • #3
Miike012 said:
Problem: When a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant:

PV = C

Q: A sample of gas is in a container at low pressure and its steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning of 10 minutes or at the end of 10 minutes?

A: dV/dt = -C/P^2(dP/dt)

By looking at my graph... at the end of 10 seconds dV/dt is approaching smaller and smaller values... thus I picked "end of 10 seconds" ... but the answer is beginning of 10 seconds... what did I do wrong?

I'm assuming this is a graph of dV/dt versus t.

At the beginning of the 10-minute interval, dV/dt is smaller (more negative) than at the end of the interval. This means that V is decreasing more at the beginning of the interval than at the end.

There is a connection between a the graph of a function and its derivative.

f' > 0 on an interval ==> f is increasing on that interval
f' < 0 on an interval ==> f is decreasing on that interval
 
  • #4
From PV= C, the pressure increases as the volume decreases. At the end of the compression, P will be greater than at the beginning so that [itex]1/P^2[/itex] will be less.
 

1. What is the relationship between pressure and volume in a sample of gas compressed at a constant temperature?

The relationship between pressure and volume in a sample of gas compressed at a constant temperature is known as Boyle's Law. This law states that the pressure of a gas is inversely proportional to its volume, meaning that as the volume decreases, the pressure increases.

2. How does compressing a gas at a constant temperature affect its density?

Compressing a gas at a constant temperature increases its density. This is because when the gas molecules are compressed into a smaller volume, they are closer together and have less space to move around, resulting in a higher concentration of molecules per unit of volume.

3. Does the temperature of a gas change when it is compressed at a constant temperature?

No, the temperature of a gas does not change when it is compressed at a constant temperature. This is because the pressure and volume of the gas are directly proportional to its temperature, according to the Ideal Gas Law. As long as the temperature remains constant, the pressure and volume will also remain constant.

4. What happens to the gas molecules when a sample of gas is compressed at a constant temperature?

When a sample of gas is compressed at a constant temperature, the gas molecules are forced closer together, resulting in an increase in pressure. The molecules do not change in size or shape, but their concentration increases, leading to a higher pressure.

5. How is the behavior of a gas at constant temperature affected by compressing it?

Compressing a gas at a constant temperature affects its behavior by increasing its pressure and density. This can also cause the gas to become more difficult to compress further, as the molecules are already packed tightly together. Additionally, the gas may exhibit more ideal gas behavior, following the Ideal Gas Law more closely, due to the constant temperature and pressure conditions.

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