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

[thirdeyechai]

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 . I appreciate any help that can be offered to me!

Thanks

 

[HallsofIvy]

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".

 

[thirdeyechai]

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

 

[sinx]

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 . I appreciate any help that can be offered to me!

Thanks

I agree with HallsofIvy. Volume is constant.

P=(R/V)dT/dt

 

[thirdeyechai]

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?

 

[sinx]

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.