Answer:
New volume = 6.0L
Explanation:
Make a list of variables given in the question, and what we want to find:
Initial volume (L) = Vi = 3.1
Initial temperature (K) = Ti = 193
New temperature (K) = Tn = 374
New volume (L) = Vn
We have volumes and temperatures given and we're talking about gas, the equation including both these variables that should come to mind is the ideal gas equation:
PV = nRT
P = pressure (Pa)
n = number of moles
R = gas constant = 8.31
Now, we construct an equation we can solve to get the new volume:
P(Vn) = nR(Tn)
And insert the values we know:
P(Vn) = n(8.31)(374)
P(Vn) = n·3107.94
The equation contains 3 unknown variables;
We should first try to see if we can eliminate the 2 we are not interested in, namely P and n;
What we should recognise is that the pressure and number of moles will not change upon warming the balloon;
Firstly, heating the gas inside the balloon doesn't add anything to the balloon, i.e. doesn't increase the moles of gas, it simply raises the energy of the gas particles already within the balloon, so n will not change;
Secondly, I think there is a mistake in the question, it should read "assume constant pressure" in the brackets, since the temperature does change, which we are told;
Also, we can assume normal atmospheric pressure inside and outside the balloon as would be the case ordinarily;
What we want to do with this information is rearrange the equation we constructed to have these 2 constant or unchanging variables on one side and everything else on the other, so:
[tex]\frac{P}{n} = \frac{3107.94}{V_{n} }[/tex]
Next, we construct an equation for the balloon before warming:
P(Vi) = nR(Ti)
P(3.1) = n(8.31)(193)
P·3.1 = n·1603.83
Once again, rearrange to get P and n on one side of the equation and everything else on the other:
[tex]\frac{P}{n} = \frac{1603.83}{3.1} \\\\ \frac{P}{n} = 517.364516[/tex]
Now, we have two equations for P/n, we can eliminate P/n since both these values remain the same before and after warming the balloon as previously established:
[tex]\frac{3107.94}{V_{n} } = \frac{P}{n} = 517.364516 \\\\ \frac{3107.94}{V_{n} } = 517.364516 \\\\[/tex]
Now, rearrange and solve for Vn:
[tex]\frac{3107.94}{V_{n} } = 517.364516 \\\\ V_{n} = \frac{3107.94}{517.364516} \\\\ V_{n} = 6.00725...[/tex]
Vn = 6.0L
