Thermal contact final temprature

[roam]

Homework Statement

The following is a worked problem from my textbook:

Consider two masses, with initial tempratures T₁ and T₂ are placed in contact and isolated from their surroundings. They are

C_V1T₁ + C_V2T₂ = C_V1T_f+C_V2T_f

This equation can be solved for the final temprature

$T_f = \frac{C_{V1}}{C_{V1}+C_{V2}} T_1 + \frac{C_{V2}}{C_{V1}+C_{V2}} T_2$

So, if I understand it correctly they have used the idea that the total internal energy is conserved, and they equated the initial to the final Q. Is that right? I'm a bit confused as how they got the expression "C_vT"? But shouldn't they be using Q=mc_vΔT? Why did they omit "m"?

The Attempt at a Solution

I can use Q=mc_vΔT and equate heat lost by the hotter mass to the heat gained by the colder mass. We don't have the mass, but I don't think we can just omit that. So, how did they get "C_vT"? I'm a bit confused. Any help is greatly appreciated.

 

[gneill]

Homework Statement

The following is a worked problem from my textbook:

Consider two masses, with initial tempratures T₁ and T₂ are placed in contact and isolated from their surroundings. They are

C_V1T₁ + C_V2T₂ = C_V1T_f+C_V2T_f

This equation can be solved for the final temprature

$T_f = \frac{C_{V1}}{C_{V1}+C_{V2}} T_1 + \frac{C_{V2}}{C_{V1}+C_{V2}} T_2$

So, if I understand it correctly they have used the idea that the total internal energy is conserved, and they equated the initial to the final Q. Is that right? I'm a bit confused as how they got the expression "C_vT"? But shouldn't they be using Q=mc_vΔT? Why did they omit "m"?

The Attempt at a Solution

I can use Q=mc_vΔT and equate heat lost by the hotter mass to the heat gained by the colder mass. We don't have the mass, but I don't think we can just omit that. So, how did they get "C_vT"? I'm a bit confused. Any help is greatly appreciated.

It's a bit like why in electronics we rate capacitors with a given capacitance rather than specify their plate area, dielectric constant, and plate separation separately; a lumped constant representing the total charge versus voltage is more convenient.

For thermal "circuits" it can be convenient to take Q=mcΔT for each fixed object or substance and turn it into Q = CT, with T now the absolute temperature and C = mc the 'heat capacitance' for the given object. It can make writing the equations for the system a bit easier if you don't have to keep dragging around individual masses and ΔT's and so forth.

The heat capacitance for a given object contains the same information as if you were given its mass and heat capacity coefficient and took the trouble to multiply them together everywhere you needed to insert them in an equation

 

[gulfcoastfella]

It's common in thermodynamics to write specific (intrinsic) quantities in lowercase and absolute (extrinsic) quantities in upper case. These kinds of measurements are referred to as intrinsic and extrinsic properties, respectively, where an intrinsic property is independent of the amount of "stuff" present (density, temperature, etc) and an extrinsic property depends on the amount of "stuff" present (weight, volume, etc).

For example, specific volume is volume per unit mass and portrayed with a lowercase v, and has the units of m$^{3}$/kg, while absolute volume is portrayed with an uppercase V, and has the units of m$^{3}$. In your case, the specific heat capacity is written as c$_{v}$ and has the units of J/(kg$\cdot$K), while the heat capacity is written as C$_{v}$ and has the units of J/K.

 

[roam]

Oh, I get it now. The small letter is the heat capacitance per unit mass so that's why the masses cancel out, and it's more convinient not to write it. Thank you very much for the help! It makes perfect sense now. :)

 

[gulfcoastfella]

Glad I could help!