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:
[tex]U_1=U_2[/tex]
The state 1 is just composed by two phases, the solid copper and the liquid nitrogen, so: [tex]U_1=U_1^C+U_1^{N_2}[/tex]
The state 2 is, by the other hand, composed by three phases, solid copper, liquid nitrogen and vapor nitrogen, so:
[tex]U_2=U_2^C+U_{2,liq}^{N_2}+U_{2,vap}^{N_2}[/tex]
So, the overall energy balance is:
[tex]U_1^C+U_1^{N_2}=U_2^C+U_{2,liq}^{N_2}+U_{2,vap}^{N_2}[/tex]
Reorganizing,
[tex]U_1^C-U_2^C=U_{2,liq}^{N_2}+U_{2,vap}^{N_2}-U_1^{N_2}[/tex]
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:
[tex]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}[/tex]
Take in mind that, for the mass balance for nitrogen, [tex]m_1^{N_2}=m_{2,liq}^{N_2}+m_{2,vap}^{N_2}[/tex],
So, let's replace [tex]m_1^{N_2}[/tex] in the energy balance:
[tex]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}[/tex]
So, as you can see, the term [tex]m_{2,liq}^{N_2}u_{2,liq}^{N_2}[/tex] disappear because [tex]u_{2,liq}^{N_2}=u_{1,liq}^{N_2}[/tex] (The specific energy in the liquid is the same because the temperature does not change).
[tex]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}[/tex]
[tex]m_C*Cp*(T_1^C-T_2^C)=m_{2,vap}^{N_2}(u_{2,vap}^{N_2}-u_1^{N_2})[/tex]
The difference [tex](u_{2,vap}^{N_2}-u_1^{N_2})[/tex] is the latent heat of vaporization because is the specific energy difference between the vapor and the liquid phases, so:
[tex]m_{2,vap}^{N_2}=\frac{m_C*Cp*(T_1^C-T_2^C)}{(u_{2,vap}^{N_2}-u_1^{N_2})}[/tex]
[tex]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[/tex]
