Question

You have two perfectly insulated cups. One contains water and the other contains an equal volume of another liquid that has half the density of water and twice the specific heat capacity. You heat the water from 10ºC to 20ºC and other liquid from 80ºC to 90ºC. Compare the amount of heat energy needed to raise the temperature of the other liquid to the amount needed to raise the temperature of the water.

Answer 1

The amount of heat absorbed is the same.

Step-by-step explanation:

Let's label water as substance 1 and the other liquid as substance 2. We know so far that:

• 
  `d_1 = 1 g/cm^3, d_2 = (1)/(2) d_1`;
• 
  `c_1 = 4.184 (J)/(g^oC), c_2 = 2c_1`;
• 
  `\Delta T_1 = \Delta T_2 = 10^oC`;
• 
  `V_1 = V_2`.

Now we know that the amount of heat absorbed is calculated by the product between specific heat capacity, mass and the change in temperature. Although the initial and final temperatures are not the same for the two substances, the change in temperature is.

Now let's express mass as a product between density and volume:

`m_1 = d_1V, m_2 = d_2V`

Heat absorbed by the water:

`Q_1 = c_1d_1V\Delta T_1`

Heat absorbed by the unknown liquid:

`Q_2 = c_2d_2V\Delta T_2 = 2c_1\cdot (1)/(2) d_1\cdot V\Delta T_1 = c_1d_1V\Delta T_1`

Notice now that:
`Q_1 = Q_2`

This means the amount of heat absorbed by each substance is the same.