What is the Thermal Conductivity of a Diatomic Gas?

[WWCY]

Homework Statement

Homework Equations

The Attempt at a Solution

Part 1.1) I managed to find the energy needed to melt the ice, before using ΔU = Nf½KΔT to solve for the new temperature, which was correct

Part 1.2) Initially tried using ΔQ = mcΔT before realising that I didn't have the mass for the diatomic gas. Am not sure how to proceed. Tried using similar logic to previous part: I set the air's reduction in internal energy ΔU = Nf½KΔT to be equal to energy gain mcΔT of the water. I then made T_final the subject of the equation. My answer turned out to be ridiculous: 269K or -3°C.

Part 2) Clueless as to how to start.

Advice would be greatly appreciated!

 

[haruspex]

: I set the air's reduction in internal energy ΔU = Nf½KΔT to be equal to energy gain mcΔT of the water. I then made Tfinal the subject of the equation. My answer turned out to be ridiculous: 269K or -3°C.

Method looks fine. No way to say where you are going wrong without seeing the details.

Part 2) Clueless as to how to start.

You are given a differential equation, but on the one side you have Q and on the other side T. You need to find another relationship between them so that you can write a D.E. Involving only one.

 

[Chestermiller]

For part 2, you are going to have to solve a time-dependent differential equation for the gas temperature as a function of time. Let $\theta$ be the temperature of the gas at time t, and let $\theta_i$ represent the temperature of the ice/water mixture. Does the temperature of this mixture change while the ice is melting? From the equation they gave you, in terms of k, $\theta$ and $\theta_i$, what is the rate of heat flow from the gas to the ice/water mixture at time t? If C is the molar heat capacity of the gas, and, knowing that you have n=2 moles of gas, what is the rate of change of internal energy of the gas with respect to time (in terms of C, n, and dT/dt)?

 

[WWCY]

For part 2, you are going to have to solve a time-dependent differential equation for the gas temperature as a function of time. Let $\theta$ be the temperature of the gas at time t, and let $\theta_i$ represent the temperature of the ice/water mixture. Does the temperature of this mixture change while the ice is melting? From the equation they gave you, in terms of k, $\theta$ and $\theta_i$, what is the rate of heat flow from the gas to the ice/water mixture at time t? If C is the molar heat capacity of the gas, and, knowing that you have n=2 moles of gas, what is the rate of change of internal energy of the gas with respect to time (in terms of C, n, and dT/dt)?

Thank you for the pointers, here's what I managed to come up with:

I let the function δT(t) = θ_g(t) - θ₀. θ_g(t) is the varying temperature of gas while θ₀ is the temperature of the melting ice which should be constant at 273.15K.

so... θ_g(t) = -(1/k) (dQ/dt) + θ₀ (I added in the minus sign to denote that heat flows from left to right)

I rationalised that dQ/dt should be equal to dU/dt of the ideal gas. As the gas loses energy, RHS of the equation below should be negative

so... ΔU = -(1/2)NfKΔT and subbing N = nN_a and f = 5, n = 2 gives ΔU = 5N_aKΔT. Taking the rate of change, dU/dt = 5N_aK(dT/dt)

subbing this back in gives:

θ_g(t) = 5N_a(dT/dt) + θ₀.

I rationalised that dT/dt actually represented the rate of change of the gas' temperature and therefore changed the equation to:

θ_g(t) = 5N_aθ_g'(t) + θ₀.

This was as far as I got, answer didnt seem to make sense though. Where did I go wrong?

Thank you in advance.

Method looks fine. No way to say where you are going wrong without seeing the details.

Did the calculations again and realized it was down to a ridiculous mistake. Apologies!

 

[Chestermiller]

Thank you for the pointers, here's what I managed to come up with:

I let the function δT(t) = θ_g(t) - θ₀. θ_g(t) is the varying temperature of gas while θ₀ is the temperature of the melting ice which should be constant at 273.15K.

so... θ_g(t) = -(1/k) (dQ/dt) + θ₀ (I added in the minus sign to denote that heat flows from left to right)

I rationalised that dQ/dt should be equal to dU/dt of the ideal gas. As the gas loses energy, RHS of the equation below should be negative

so... ΔU = -(1/2)NfKΔT and subbing N = nN_a and f = 5, n = 2 gives ΔU = 5N_aKΔT. Taking the rate of change, dU/dt = 5N_aK(dT/dt)

subbing this back in gives:

θ_g(t) = 5N_a(dT/dt) + θ₀.

I rationalised that dT/dt actually represented the rate of change of the gas' temperature and therefore changed the equation to:

θ_g(t) = 5N_aθ_g'(t) + θ₀.

This was as far as I got, answer didnt seem to make sense though. Where did I go wrong?

Thank you in advance.
Did the calculations again and realized it was down to a ridiculous mistake. Apologies!

You seem to have canceled the two k's even though they are not the same parameter. And, you have a sign error.

 

[WWCY]

You seem to have canceled the two k's even though they are not the same parameter. And, you have a sign error.

i changed θg(t) = -(1/k) (dQ/dt) + θ₀ back to θg(t) = (1/k) (dQ/dt) + θ₀ while keeping ΔU = -(1/2)NfKΔT the same,

I ended up with an equation:

θ_g(t) = -(5NaK_b / K) θ_g'(t) + θ₀.

Does this look right?

 

[Chestermiller]

I get $$5R\frac{d\theta_g}{dt}=-k(\theta_g-\theta_0)$$where R is the universal gas constant. Do you know how to solve this differential equation?

 

[WWCY]

I get $$5R\frac{d\theta_g}{dt}=-k(\theta_g-\theta_0)$$where R is the universal gas constant. Do you know how to solve this differential equation?

Yes, thank you! However solving it with either Kelvin or Celcius (I believe either should work) gives me a time of about 5 seconds, a far cry from the 20s in the answer.

My final equation was:

θ_g = θ + C/e^(k/5R)t

and I changed k to 2.093J/(s°C) rather than leaving it in cal.

Is this correct? Thank you so much for your patience, greatly appreciate it.

 

[Chestermiller]

I get 5 sec also.