Answer:
1.2584kg of nitrogen boils.
Explanation:
Consider the energy balance for the overall process. There are not heat or work fluxes to the system, so the total energy keeps the same.
For the explanation, the 1 and 2 subscripts will mean initial and final state, and C and N2 superscripts will mean copper and nitrogen respectively; also, liq and vap will mean liquid and vapor phase respectively.
The overall energy balance for the whole system is:
![U_1=U_2](https://tex.z-dn.net/?f=U_1%3DU_2)
The state 1 is just composed by two phases, the solid copper and the liquid nitrogen, so: ![U_1=U_1^C+U_1^{N_2}](https://tex.z-dn.net/?f=U_1%3DU_1%5EC%2BU_1%5E%7BN_2%7D)
The state 2 is, by the other hand, composed by three phases, solid copper, liquid nitrogen and vapor nitrogen, so:
![U_2=U_2^C+U_{2,liq}^{N_2}+U_{2,vap}^{N_2}](https://tex.z-dn.net/?f=U_2%3DU_2%5EC%2BU_%7B2%2Cliq%7D%5E%7BN_2%7D%2BU_%7B2%2Cvap%7D%5E%7BN_2%7D)
So, the overall energy balance is:
![U_1^C+U_1^{N_2}=U_2^C+U_{2,liq}^{N_2}+U_{2,vap}^{N_2}](https://tex.z-dn.net/?f=U_1%5EC%2BU_1%5E%7BN_2%7D%3DU_2%5EC%2BU_%7B2%2Cliq%7D%5E%7BN_2%7D%2BU_%7B2%2Cvap%7D%5E%7BN_2%7D)
Reorganizing,
![U_1^C-U_2^C=U_{2,liq}^{N_2}+U_{2,vap}^{N_2}-U_1^{N_2}](https://tex.z-dn.net/?f=U_1%5EC-U_2%5EC%3DU_%7B2%2Cliq%7D%5E%7BN_2%7D%2BU_%7B2%2Cvap%7D%5E%7BN_2%7D-U_1%5E%7BN_2%7D)
The left part of the equation can be written in terms of the copper Cp because for solids and liquids Cp≅Cv. The right part of the equation is written in terms of masses and specific internal energy:
![m_C*Cp*(T_1^C-T_2^C)=m_{2,liq}^{N_2}u_{2,liq}^{N_2}+m_{2,vap}^{N_2}u_{2,vap}^{N_2}-m_1^{N_2}u_1^{N_2}](https://tex.z-dn.net/?f=m_C%2ACp%2A%28T_1%5EC-T_2%5EC%29%3Dm_%7B2%2Cliq%7D%5E%7BN_2%7Du_%7B2%2Cliq%7D%5E%7BN_2%7D%2Bm_%7B2%2Cvap%7D%5E%7BN_2%7Du_%7B2%2Cvap%7D%5E%7BN_2%7D-m_1%5E%7BN_2%7Du_1%5E%7BN_2%7D)
Take in mind that, for the mass balance for nitrogen,
,
So, let's replace
in the energy balance:
![m_C*Cp*(T_1^C-T_2^C)=m_{2,liq}^{N_2}u_{2,liq}^{N_2}+m_{2,vap}^{N_2}u_{2,vap}^{N_2}-m_{2,liq}^{N_2}u_1^{N_2}-m_{2,vap}^{N_2}u_1^{N_2}](https://tex.z-dn.net/?f=m_C%2ACp%2A%28T_1%5EC-T_2%5EC%29%3Dm_%7B2%2Cliq%7D%5E%7BN_2%7Du_%7B2%2Cliq%7D%5E%7BN_2%7D%2Bm_%7B2%2Cvap%7D%5E%7BN_2%7Du_%7B2%2Cvap%7D%5E%7BN_2%7D-m_%7B2%2Cliq%7D%5E%7BN_2%7Du_1%5E%7BN_2%7D-m_%7B2%2Cvap%7D%5E%7BN_2%7Du_1%5E%7BN_2%7D)
So, as you can see, the term
disappear because
(The specific energy in the liquid is the same because the temperature does not change).
![m_C*Cp*(T_1^C-T_2^C)=m_{2,vap}^{N_2}u_{2,vap}^{N_2}-m_{2,vap}^{N_2}u_1^{N_2}](https://tex.z-dn.net/?f=m_C%2ACp%2A%28T_1%5EC-T_2%5EC%29%3Dm_%7B2%2Cvap%7D%5E%7BN_2%7Du_%7B2%2Cvap%7D%5E%7BN_2%7D-m_%7B2%2Cvap%7D%5E%7BN_2%7Du_1%5E%7BN_2%7D)
![m_C*Cp*(T_1^C-T_2^C)=m_{2,vap}^{N_2}(u_{2,vap}^{N_2}-u_1^{N_2})](https://tex.z-dn.net/?f=m_C%2ACp%2A%28T_1%5EC-T_2%5EC%29%3Dm_%7B2%2Cvap%7D%5E%7BN_2%7D%28u_%7B2%2Cvap%7D%5E%7BN_2%7D-u_1%5E%7BN_2%7D%29)
The difference
is the latent heat of vaporization because is the specific energy difference between the vapor and the liquid phases, so:
![m_{2,vap}^{N_2}=\frac{m_C*Cp*(T_1^C-T_2^C)}{(u_{2,vap}^{N_2}-u_1^{N_2})}](https://tex.z-dn.net/?f=m_%7B2%2Cvap%7D%5E%7BN_2%7D%3D%5Cfrac%7Bm_C%2ACp%2A%28T_1%5EC-T_2%5EC%29%7D%7B%28u_%7B2%2Cvap%7D%5E%7BN_2%7D-u_1%5E%7BN_2%7D%29%7D)
![m_{2,vap}^{N_2}=\frac{3kg*0.092\frac{cal}{gC} *(296.15K-77.3K)}{48.0\frac{cal}{g}}\\m_{2,vap}^{N_2}=1.2584kg](https://tex.z-dn.net/?f=m_%7B2%2Cvap%7D%5E%7BN_2%7D%3D%5Cfrac%7B3kg%2A0.092%5Cfrac%7Bcal%7D%7BgC%7D%20%2A%28296.15K-77.3K%29%7D%7B48.0%5Cfrac%7Bcal%7D%7Bg%7D%7D%5C%5Cm_%7B2%2Cvap%7D%5E%7BN_2%7D%3D1.2584kg)