A 226 g sample of brass at 105◦C is placed in
a calorimeter cup that contains 283 g of water
at 29◦C.
If you disregard the absorption of heat
by the cup, what will be the final temperature? Assume the specific heat of brass is 376 J/kg ·◦ C, and the specific heat of water is 4180 J/kg ·◦ C.
Answer in units of ◦C.
heat lost by brass is gained by water
226 * 376 * (105 - t) = 283 * 4180 * (t - 29)
solve for t
To find the final temperature, we can use the principle of conservation of energy. The heat lost by the brass will be equal to the heat gained by the water.
The heat lost by the brass can be calculated using the equation:
Q_lost = m_brass * c_brass * (T_final - T_initial)
Where:
m_brass = mass of brass = 0.226 kg
c_brass = specific heat of brass = 376 J/kg ·◦ C
T_initial = initial temperature of brass = 105◦C
The heat gained by the water can be calculated using the equation:
Q_gained = m_water * c_water * (T_final - T_initial)
Where:
m_water = mass of water = 0.283 kg
c_water = specific heat of water = 4180 J/kg ·◦ C
T_initial = initial temperature of water = 29◦C
Since the two quantities are equal, we can set them equal to each other and solve for T_final:
m_brass * c_brass * (T_final - T_initial) = m_water * c_water * (T_final - T_initial)
(0.226 kg) * (376 J/kg ·◦ C) * (T_final - 105◦C) = (0.283 kg) * (4180 J/kg ·◦ C) * (T_final - 29◦C)
Simplifying the equation:
85.376 * (T_final - 105) = 1181.94 * (T_final - 29)
Multiplying out:
85.376 * T_final - 8962.88 = 1181.94 * T_final - 34229.26
Now, collecting the terms with T_final on one side:
85.376 * T_final - 1181.94 * T_final = -34229.26 + 8962.88
-1096.564 * T_final = -25266.38
Now, dividing both sides by -1096.564:
T_final = (-25266.38) / (-1096.564)
T_final ≈ 23.04 ◦C
Therefore, the final temperature will be approximately 23.04 ◦C.
To find the final temperature, we need to use the principle of energy conservation. The heat lost by the brass will be equal to the heat gained by the water.
The heat lost by the brass can be calculated using the formula:
Q_lost = m_brass * c_brass * (T_initial - T_final)
Where:
Q_lost is the heat lost by the brass
m_brass is the mass of the brass (226 g)
c_brass is the specific heat of brass (376 J/kg ·◦ C)
T_initial is the initial temperature of the brass (105◦C)
T_final is the final temperature we want to find
The heat gained by the water can be calculated using the formula:
Q_gained = m_water * c_water * (T_final - T_initial_water)
Where:
Q_gained is the heat gained by the water
m_water is the mass of the water (283 g)
c_water is the specific heat of water (4180 J/kg ·◦ C)
T_initial_water is the initial temperature of the water (29◦C)
Since the heat lost by the brass is equal to the heat gained by the water, we have:
Q_lost = Q_gained
Plugging in the values:
m_brass * c_brass * (T_initial - T_final) = m_water * c_water * (T_final - T_initial_water)
Simplifying the equation:
226 * 376 * (105 - T_final) = 283 * 4180 * (T_final - 29)
Now, we can solve this equation to find the value of T_final.