Question

What is the final equilibrium temperature when 22 g of milk at 9 °C is added to 143 g of coffee at 98 °C? Assume the specific heats of milk and coffee are the same as that of water, and neglect the specific heat of the container. Provide the answer in units of °C.

Answer 1

The final equilibrium temperature is calculated using the conservation of energy, where the heat lost by the hot coffee equals the heat gained by the cold milk, assuming no heat loss to the environment.

Step-by-step explanation:

To find the final equilibrium temperature T_f when two substances at different temperatures are mixed, without heat loss to the environment or container, we use the concept of conservation of energy. Specifically, the heat lost by the hotter substance must equal the heat gained by the colder substance, assuming the substances have the same specific heat capacity.

The equation for the heat Q gained or lost by a substance is Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. Applying conservation of energy, m_coffee * c * (T_f - T_initial_coffee) = m_milk * c * (T_initial_milk - T_f), where m_coffee is the mass of coffee, m_milk is the mass of milk, T_initial_coffee is the initial temperature of the coffee, and T_initial_milk is the initial temperature of the milk.

We are given: m_coffee = 143g, m_milk = 22g, T_initial_coffee = 98°C, T_initial_milk = 9°C, and the specific heat capacity c for both coffee and milk is the same as water, which is 4.18 J/g°C. Plugging in these values and solving for T_f, the final temperature can be calculated. As both the milk and the coffee share water's specific heat, the units of grams and degrees Celsius will cancel out in the algebra, making it unnecessary to convert to moles or any other unit.