a) Object A
When a certain amount of energy Q is supplied to a sample of substance with mass m, the temperature of the substance increases by , according to the equation
:
where is the specific heat capacity of the substance
.
The equation can be rewritten as
In our problem, we have two different objecs A and B (we assume they have same mass and they are at same initial temperature). Both objects are placed in a beaker containing 1000 g of water at 10.0 °C. Each object gives off energy to the water, until the object is in thermal equilibrium (=same temperature) with the water. The amount of energy given off by each object is equal to the amount of energy absorbed by the water, so:
where
mw is the mass of water
Cw is the specifi heat of water
Teq is the temperature at equilibrium
Twi is the initial temperature of water
is the mass of the object
is the specific heat capacity of the object
is the initial temperature of the object
Solving for ,
For object A, the increase in temperature of the water is
So the formula becomes
For object B, the increase in temperature of the water is
So the formula becomes
Also, we have to notice that if the two objects start at the same temperature, then the equilibrium temperature in case A) is higher than in case B, therefore the denominator is lower for case A than case B: this means that overall, the specific heat capacity of object A must be larger than that of object B.
b)
We can also give a quantitative comparison of the two specific heat capacities.
For object A we have:
For object B:
Also, we know that the equilibrium temperature for object A is
higher than in case B, so we can write the second equation as
Now we can calculate the ratio of the two specific heat capacities: