Ideal Gas Law Related Rates: The ideal gas constant R ≈ 8.3145 kPaL/Kmol

thirdeyechai

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It's my first time trying to work through real world applications of calculus and I'm having a hard time with it. Below is the question I'm working on:

Given the Ideal Gas Law PV = nRT where:
P - pressure (kPa)
V - volume (L)
n - moles of gas
R - ideal gas constant
T - temperature (K)

The ideal gas constant R ≈ 8.3145 kPaL/Kmol

Suppose that one mole of ideal gas is held in a closed container with a volume of25 litres. If the temperature of the gas is increased at a rate of 3.5 kelvin/min, how quicklywill the pressure increase?

Now the question gives us dT/dt = 3.5 kelvin/min and is asking us to find dP/dt in kPa/min. We also know V = 25 and n =1

As n = 1 we can re-write the original equation as: PV = RT. I guess my problem is it seems like there are too many unknowns. I would perhaps start by finding the derivative of PV = RT:

P·dV/dt + V·dP/dt = R·dT/dt + T·dR/dt (Using the Product Rule)

As dR/dt = 0 (derivative of any constant is zero) I could simply the equation to:

P·dV/dt + V·dP/dt = R·dT/dt

I would also try and rearrange the equation to solve for dP/dt

V·dP/dt = R·dT/dt - P·dV/dt

(R·dT/dt - P·dV/dt)
dP/dt = -----------------------
V

(R·dT/dt - P·dV/dt)
dP/dt = -----------------------
V

I can then input values V, dT/dt and R

(8.3145·3.5 k/m - P·dV/dt)
dP/dt = -----------------------
25

But from here I'm lost (I'm not even sure if I'm on the right track to begin with tbh). Do I need to somehow find the value of dV/dt before I continue? Just feeling lost in general with this one :cry:. I appreciate any help that can be offered to me!

Thanks
 
You can't find \(\displaystyle \frac{dV}{dt}\)! Unless you have other information, V is independent of P and T.


If, for example, this is all happening inside a vessel with rigid walls (glass or metal) so that the volume cannot change, then \(\displaystyle \frac{dV}{dt}= 0\).
Or this could be an "adiabatic" expansion in which no heat is gained or lost. In that case \(\displaystyle PV^\gamma\)= constant where \(\displaystyle gamma\) is the "specific heat at constant pressure" divided by the "specific heat at constant volume".
 
You can't find \(\displaystyle \frac{dV}{dt}\)! Unless you have other information, V is independent of P and T.

If, for example, this is all happening inside a vessel with rigid walls (glass or metal) so that the volume cannot change, then \(\displaystyle \frac{dV}{dt}= 0\).
Or this could be an "adiabatic" expansion in which no heat is gained or lost. In that case \(\displaystyle PV^\gamma\)= constant where \(\displaystyle gamma\) is the "specific heat at constant pressure" divided by the "specific heat at constant volume".

Right, I should mention this is taking place in a closed container
 
It's my first time trying to work through real world applications of calculus and I'm having a hard time with it. Below is the question I'm working on:

Given the Ideal Gas Law PV = nRT where:
P - pressure (kPa)
V - volume (L)
n - moles of gas
R - ideal gas constant
T - temperature (K)

The ideal gas constant R ≈ 8.3145 kPaL/Kmol

Suppose that one mole of ideal gas is held in a closed container with a volume of25 litres. If the temperature of the gas is increased at a rate of 3.5 kelvin/min, how quicklywill the pressure increase?

Now the question gives us dT/dt = 3.5 kelvin/min and is asking us to find dP/dt in kPa/min. We also know V = 25 and n =1

As n = 1 we can re-write the original equation as: PV = RT. I guess my problem is it seems like there are too many unknowns. I would perhaps start by finding the derivative of PV = RT:

P·dV/dt + V·dP/dt = R·dT/dt + T·dR/dt (Using the Product Rule)

As dR/dt = 0 (derivative of any constant is zero) I could simply the equation to:

P·dV/dt + V·dP/dt = R·dT/dt

I would also try and rearrange the equation to solve for dP/dt

V·dP/dt = R·dT/dt - P·dV/dt

(R·dT/dt - P·dV/dt)
dP/dt = -----------------------
V

(R·dT/dt - P·dV/dt)
dP/dt = -----------------------
V

I can then input values V, dT/dt and R

(8.3145·3.5 k/m - P·dV/dt)
dP/dt = -----------------------
25

But from here I'm lost (I'm not even sure if I'm on the right track to begin with tbh). Do I need to somehow find the value of dV/dt before I continue? Just feeling lost in general with this one :cry:. I appreciate any help that can be offered to me!

Thanks

I agree with HallsofIvy. Volume is constant.

P=(R/V)dT/dt
 
Thanks for the responses guy! So my working from here has been as follows:

Finding the derivative of PV = RT (as n = 1 we can omit it)

P·dV/dt + V·dP/dt = R(dT/dt)

25·dP/dt = 8.3145 · 3.5 (as dV/dt = 0)

dP/dt = 1.16403 kPa/min

So as the temperature is increased at a rate of 3.5 kelvin/min, the pressure increases at a rate of 1.16403 kPa/min.

Part 2 of the question is asking for when temperature is a constant 300K and the volume is decreasing at a rate of 2.0L/min. How quickly is the pressure of the gas increasing at the instant when V = 20. So this time temperature is the constant so dT/dt will be zero. I'm just wondering what is your methodology when answering questions like this? Is there any particular way the equation should be rearranged to solve for this?
 
Thanks for the responses guy! So my working from here has been as follows:

Finding the derivative of PV = RT (as n = 1 we can omit it)

P·dV/dt + V·dP/dt = R(dT/dt)

25·dP/dt = 8.3145 · 3.5 (as dV/dt = 0)

dP/dt = 1.16403 kPa/min

So as the temperature is increased at a rate of 3.5 kelvin/min, the pressure increases at a rate of 1.16403 kPa/min.

Part 2 of the question is asking for when temperature is a constant 300K and the volume is decreasing at a rate of 2.0L/min. How quickly is the pressure of the gas increasing at the instant when V = 20. So this time temperature is the constant so dT/dt will be zero. I'm just wondering what is your methodology when answering questions like this? Is there any particular way the equation should be rearranged to solve for this?

PV=nRT in both cases (all cases involving ideal gas law).
For both questions, solve the equation for P, i.e. P=[nRT]/V.
taking the derivative of both sides, with respect to V or T, gives you either dP/dV, or dP/dT.
then dP/dt=dP/dV*dV/dt;
or dP/dt=dP/dT*dT/dt.
 
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