Consider a glass of 294 mL of water at 29°C. Calculate the mass of ice at -15°C that must be added to cool the water to 10°C after thermal equilibrium is achieved

To calculate the mass of ice needed to cool the water, we need to use the principle of heat exchange and the specific heat capacities of water and ice.

The basic idea is that the heat lost by the water will equal the heat gained by the ice when they reach thermal equilibrium.

Here are the steps to calculate the mass of ice:

Step 1: Calculate the heat lost by the water.
Using the formula Q = mcΔT, where Q is the heat lost, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature, we can determine the heat lost by the water.

Given:
m(water) = 294 mL = 294 g (since the density of water is approximately 1 g/mL)
c(water) = 4.18 J/g°C (specific heat capacity of water)
ΔT(water) = T(final) - T(initial) = 10°C - 29°C = -19°C

Using the formula Q = mcΔT, we have:
Q(water) = m(water) * c(water) * ΔT(water)

Step 2: Calculate the heat gained by the ice.
Using the same formula, Q = mcΔT, we can calculate the heat gained by the ice.

Given:
c(ice) = 2.09 J/g°C (specific heat capacity of ice)
ΔT(ice) = T(final) - T(initial) = 10°C - (-15°C) = 25°C (the change in temperature is always positive)

Using the formula Q = mcΔT, we have:
Q(ice) = m(ice) * c(ice) * ΔT(ice)

Step 3: Equate the heat lost by the water to the heat gained by the ice.
Since the heat lost by the water is equal to the heat gained by the ice, we can set up the following equation:

Q(water) = Q(ice)
m(water) * c(water) * ΔT(water) = m(ice) * c(ice) * ΔT(ice)

Step 4: Solve for the mass of ice (m(ice)).
Rearranging the equation, we get:
m(ice) = (m(water) * c(water) * ΔT(water)) / (c(ice) * ΔT(ice))

Plugging in the given values and solving the equation will give us the mass of ice required.

Note: Be sure to convert the values to a consistent unit system, either all in grams or all in kilograms, before calculating.