A precious-stone dealer wishes to find the specific heat capacity of a 0.036-kg gemstone. The specimen is heated to 95.0°C and then placed in a 0.17-kg copper vessel that contains 0.086 kg of water at equilibrium at 25.0°C. The loss of heat to the external environment is negligible. When equilibrium is established, the temperature is 29.5°C. What is the specific heat capacity of the specimen?
To find the specific heat capacity of the gemstone, we can use the principle of energy conservation.
The energy gained by the gemstone should be equal to the sum of the energy lost by the gemstone and the energy absorbed by the copper vessel and the water.
Here are the steps to solve the problem:
Step 1: Calculate the energy gained by the gemstone:
The energy gained by the gemstone can be calculated using the formula:
Q1 = mcΔT, where
Q1 is the energy gained by the gemstone,
m is the mass of the gemstone (0.036 kg),
c is the specific heat capacity of the gemstone (unknown),
and ΔT is the change in temperature of the gemstone (95.0°C - 29.5°C).
Step 2: Calculate the energy lost by the gemstone:
The energy lost by the gemstone can be calculated using the formula:
Q2 = mcΔT, where
Q2 is the energy lost by the gemstone,
m is the mass of the gemstone (0.036 kg),
c is the specific heat capacity of the gemstone (unknown),
and ΔT is the change in temperature of the gemstone (initial temperature - final temperature).
Step 3: Calculate the energy absorbed by the copper vessel:
The energy absorbed by the copper vessel can be calculated using the formula:
Q3 = mcΔT, where
Q3 is the energy absorbed by the copper vessel,
m is the mass of the copper vessel (0.17 kg),
c is the specific heat capacity of copper (which is 387 J/kg°C),
and ΔT is the change in temperature of the copper vessel (initial temperature - final temperature).
Step 4: Calculate the energy absorbed by the water:
The energy absorbed by the water can be calculated using the formula:
Q4 = mcΔT, where
Q4 is the energy absorbed by the water,
m is the mass of the water (0.086 kg),
c is the specific heat capacity of water (which is 4186 J/kg°C),
and ΔT is the change in temperature of the water (initial temperature - final temperature).
Step 5: Set up the energy conservation equation:
Since the total energy gained by the gemstone is equal to the sum of the energies lost by the gemstone, absorbed by the copper vessel, and absorbed by the water, we can write the equation:
Q1 = Q2 + Q3 + Q4
Step 6: Plug in the values and solve for the specific heat capacity of the gemstone:
Substitute the known values into the equation, rearrange it to solve for c, and calculate the specific heat capacity of the gemstone.