Derivation

Time to turn up the spiciness just a little bit! This next example is similar to the previous one in that you will only use basic arithmetic operations. However, there are a lot

more variables in this problem, and some mathematical substitutions will be required. But that's okay! No matter what you call a variable, it works the exact same way as

any other variable. You can probably deduce the answers after one attempt, but make sure you can actually derive each expression! You will be expected to do this kind of

math in this class.

Consider an isolated system composed of two thermodynamic system components, A and $B,$ in thermal contact and each at different initial temperatures and $\{$:$T_{i,B}).$ The

nonzero heat $(q)$ transferred into or out of each system can be related across the systems by:

$q_A=-q_B$

where each heat transfer can be expressed as:

$q_A=m_A*C_{s,A}*T_A$ and $q_B=m_B*C_{s,B}*T_B$

In the above expressions, $m_x$ is the mass of system component $x($where $x=A$ or $B),C_{s,x}$ is the specific heat capacity of component $x,$ and $T_x$ is the temperature change

from the initial conditions to thermal equilibrium for component $x.$ Note that $T_x=T_{f,x}-T_{i,x}$ where $f$ and i indicate the final and initial temperatures of component $x,$

respectively.

Label each of the following mathematical statements as true or false given $q_A=-q_B,q_x=m_x*C_{s,x}*T_x,$ and $T_x=T_{f,x}-T_{i,x}$ are true.

$(1)m_A*C_{s,A}*T_A=-m_B*C_{s,B}*T_B$

$(2)q_A=-m_B*C_{s,B}*T_B$

$(3)m_A*C_{s,A}*(T_{f,A}-T_{i,A})=-m_B*C_{s,B}*(T_{f,B}-T_{i,B})$

choose your answer...

$(4)-T_A*C_{s,A}*m_A=C_{s,B}*m_B*T_B$

choose your answer...

$(5)m_A*C_{s,A}*T_{f,A}-m_A*C_{s,A}*T_{i,A}=-m_B*C_{s,B}*T_{f,B}+m_B*C_{s,B}*T_{i,B}$

choose your answer...

$(6)m_A*C_{s,A}*T_{f,A}+m_B*C_{s,B}*T_{f,B}=m_A*C_{s,A}*T_{i,A}+m_B*C_{s,B}*T_{i,B}$

choose your answer...