How to Calculate Ideal Gas Temperature Change without Bursting the Vessel

[AdkinsJr]

Homework Statement

A certain amount of gas at 298.15 K and at a pressure of 0.800 atm is contained in a glass vessel. suppose that the vessel can withstand a pressure of 2.00 atm. How high can you raise the termperature of the gas without bursting the vessel.

Homework Equations

$$PV=nRT$$

Since the amount of gas remains constant:

$$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

The Attempt at a Solution

Obviously I don't have enough information to fill this in directly. Ideal gas problems are usually very simple, but this one stumped me. I don't see how I can solve for $$T_2$$

 

[CompuChip]

You are almost there.
What can you say about the relation between V₁ and V₂?

 

[AdkinsJr]

You are almost there.
What can you say about the relation between V₁ and V₂?

Thanks for your reply. I noticed that they must be equal right? I kind of forgot the basic definition of a gas, lol. The volume must be constant if the gas has filled the container.

 

[Borek]

Yep, it is just pV=const.

Beware:

$$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

This is incorrect. It would be correct for exactly 1 mole of gas.

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methods

 

[AdkinsJr]

Yep, it is just pV=const.

Beware:

$$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

This is incorrect. It would be correct for exactly 1 mole of gas.

--
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I don't understand why. Isn't $$n$$ a constant? The formula is in my book.

 

[Borek]

No, n is number of moles. It is usually constant throughout the problem if the number of moles of gas doesn't change, but it is not constant as R is.

Ideal gas equation is

PV=nRT

That means

$$R = \frac{PV}{nT}$$

or

$$nR = \frac{PV}{T}$$

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methods

 

[AdkinsJr]

No, n is number of moles. It is usually constant throughout the problem if the number of moles of gas doesn't change, but it is not constant as R is.

Ideal gas equation is

PV=nRT

That means

$$R = \frac{PV}{nT}$$

or

$$nR = \frac{PV}{T}$$

--
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Yeah, I know that n isn't a natural constant. I meant that it's constant in this particular problem since it doesn't change. The number of moles before and after are equal.

$$\frac{V_1P_1}{n_1T_1}=\frac{V_2P_2}{n_2T_2}$$

$$n_1=n_2=n$$

$$\frac{V_1P_1}{nT_1}=\frac{V_2P_2}{nT_2}$$

$$\frac{V_1P_1}{T_1}=\frac{V_2P_2}{T_2}$$

 

[Borek]

OK, but still

$$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

is incorrect in general (holds only for one mole of gas). It should be

$$nR=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

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[CompuChip]

So in this case, you could continue by writing V₁ = V₂ = V, so
$$\frac{n R}{V} = \frac{P_1}{T_1} = \frac{P_2}{T_2}$$
where you can call the left hand side R', or C, or k (not k_B :) ), since it is a constant in the current problem.

You could calculate the constant if you knew n and V and looked up R. However, the equality in that formula which you are interested in, is of course the second one:
$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

 

[AdkinsJr]

OK, but still

$$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

is incorrect in general (holds only for one mole of gas). It should be

$$nR=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

--
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Oh, ok, I see what you're saying. I wasn't thinking when I wrote $$R=\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$. I agree that this is incorrect. I thought that you were claiming that $$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$ was incorrect.

Thanks