To answer these questions, we need to use the concept of specific heat capacity and the formula for heat transfer.
(a) First, we need to calculate the heat required to raise the temperature of the ice block to its final temperature.
The formula for heat transfer is given by:
Q = m * c * ΔT
Where:
Q is the heat transfer (in Joules),
m is the mass of the object (in kilograms),
c is the specific heat capacity (in J/(kg·°C)),
ΔT is the change in temperature (in °C).
For ice, the specific heat capacity is approximately 2090 J/(kg·°C).
Given:
m = 1.2 kg (mass of the ice block)
c = 2090 J/(kg·°C) (specific heat capacity of ice)
ΔT = final temperature - initial temperature = final temperature - (-5°C) = final temperature + 5°C
We also know that Q = 5.8 * 10^5 J (given heat transfer).
Substituting the known values into the formula:
5.8 * 10^5 J = 1.2 kg * 2090 J/(kg·°C) * (final temperature + 5°C)
Now let's solve for the final temperature:
5.8 * 10^5 J = (1.2 kg * 2090 J/(kg·°C)) * (final temperature + 5°C)
Divide both sides by (1.2 kg * 2090 J/(kg·°C)):
(final temperature + 5°C) = (5.8 * 10^5 J) / (1.2 kg * 2090 J/(kg·°C))
(final temperature + 5°C) = 23.79°C
Subtract 5°C from both sides:
final temperature = 23.79°C - 5°C
(final temperature) = 18.79°C
So, the final temperature of the ice block is 18.79°C.
(b) Next, we need to determine the factor by which the mass of the ice block needs to be increased if the same final temperature is to be achieved with an increased heat transfer.
Let's call the increased heat transfer factor "F".
We know the mass of the ice block is initially 1.2 kg.
To achieve the same final temperature, the new mass of the ice block can be calculated using the formula for heat transfer:
Q = F * m_new * c * ΔT
where m_new is the new mass of the ice block.
Since the final temperature and specific heat capacity remain constant, the equation becomes:
F * m_new = Q / (c * ΔT)
We know that F = 3.0 (given increase in heat transfer), and we can substitute the other known values:
3.0 * m_new = (5.8 * 10^5 J) / (1.2 kg * 2090 J/(kg·°C) * 18.79°C)
Simplify the equation:
m_new = (5.8 * 10^5 J) / (1.2 kg * 2090 J/(kg·°C) * 18.79°C * 3.0)
m_new = 1.6066416 kg
Therefore, the mass of the ice block must be increased by a factor of 1.6066416 / 1.2 ≈ 1.339 for the system to have the same final temperature.
To summarize:
(a) The final temperature of the ice block is approximately 18.79°C.
(b) The mass of the ice block must be increased by a factor of approximately 1.339 for the system to have the same final temperature when the heat transfer is increased by a factor of 3.0.