Substances A and B, initially at different temperatures, come in contact with each other and reach
thermal equilibrium. The mass of substance A is twice the mass of substance B. The specifi c
heat capacity of substance B is twice the specifi c heat capacity of substance A. Which statement
is true about the fi nal temperature of the two substances once thermal equilibrium is reached?
(a) The fi nal temperature will be closer to the initial temperature of substance A than substance B.
(b) The fi nal temperature will be closer to the initial temperature of substance B than substance A.
(c) The fi nal temperature will be exactly midway between the initial temperatures of substances
A and B.
If you have trouble with a problem like this with "reasoning it out", I suggest you assign numbers. Perhaps something like this.
Say A = 100 g and is twice B which makes B 50 g. Check that.
Specific heat A = 1. B is 2x A so B is 2. Check that.
Initial T A = 100; initial B = 50
[mass A x specific heat A x (Tfinal-Tinital)] + [mass B x specific heat B x (Tfinal-Tinitial)] = 0
Solve for Tfinal and check that against the answers.
Then the fun of it change Tinitial for A = 50 and Tinitial B is 100. See if your conclusion is the same for either setting.
To determine the final temperature, we need to consider the conservation of energy principle. When substance A and substance B come in contact and reach thermal equilibrium, the heat lost by substance A must be equal to the heat gained by substance B.
Now, let's use some variables to solve this problem:
- Let Ta be the initial temperature of substance A.
- Let Tb be the initial temperature of substance B.
- Let Ma be the mass of substance A.
- Let Mb be the mass of substance B.
- Let Ca be the specific heat capacity of substance A.
- Let Cb be the specific heat capacity of substance B.
- Let Tf be the final temperature of substances A and B.
According to the conservation of energy principle:
(Qlost by A) = (Qgained by B)
The heat lost by substance A can be calculated using the formula:
(Qlost by A) = (Ma) * (Ca) * (Ta - Tf)
The heat gained by substance B can be calculated using the formula:
(Qgained by B) = (Mb) * (Cb) * (Tf - Tb)
Since these two quantities are equal, we can set up the equation:
(Ma) * (Ca) * (Ta - Tf) = (Mb) * (Cb) * (Tf - Tb)
Given that Ma = 2Mb and Cb = 2Ca, we can substitute these values into the equation:
(2Mb) * (Ca) * (Ta - Tf) = (Mb) * (2Ca) * (Tf - Tb)
Simplifying the equation:
(Ta - Tf) = 2(Tf - Tb)
Expanding and rearranging:
Ta - Tf = 2Tf - 2Tb
3Tf = Ta + 2Tb
Tf = (1/3)(Ta + 2Tb)
From this equation, we see that Tf is a weighted average of Ta and Tb. Since the mass ratio is 2:1, and the specific heat capacity ratio is 1:2, the final temperature Tf will be closer to the initial temperature of substance B than substance A.
Therefore, the correct statement is: (b) The final temperature will be closer to the initial temperature of substance B than substance A.
The final temperature of two substances in thermal equilibrium can be determined using the principle of conservation of energy.
First, let's define some variables:
T_A = initial temperature of substance A
T_B = initial temperature of substance B
m_A = mass of substance A
m_B = mass of substance B
c_A = specific heat capacity of substance A
c_B = specific heat capacity of substance B
T_f = final temperature of both substances
According to the principle of conservation of energy, the heat lost by substance A equals the heat gained by substance B:
m_A * c_A * (T_f - T_A) = m_B * c_B * (T_f - T_B)
Given that m_A = 2m_B and c_B = 2c_A, we can rewrite the equation as:
2m_B * c_A * (T_f - T_A) = m_B * 2c_A * (T_f - T_B)
Simplifying the equation:
2(T_f - T_A) = (T_f - T_B)
Expanding:
2T_f - 2T_A = T_f - T_B
Rearranging the terms:
T_f = 2T_A - T_B
From the equation, we can see that the final temperature is a weighted average of the initial temperatures, with a weight of 2 for T_A and -1 for T_B.
Therefore, the correct answer is (a) The final temperature will be closer to the initial temperature of substance A than substance B.