When ice at 0°C melts to liquid water at 0°C, it absorbs 0.334 kJ of heat per gram. Suppose the heat needed to melt 30.7 g of ice is absorbed from the water contained in a glass. If this water has a mass of 0.189 kg and a temperature of 21.0°C, what is the final temperature of the water? (Note that you will also have 30.7 g of water at 0°C from the ice.)

The final temperature of the water is 8.3°C.

To find the final temperature of the water, we can apply the principle of conservation of energy. The heat absorbed by the ice is equal to the heat lost by the water.

The heat absorbed by the ice (Q_ice) can be calculated using the equation:

Q_ice = mass_ice * heat_capacity_ice * temperature_change_ice

Given that the heat absorbed by the ice is 0.334 kJ/g (or 334 J/g), and the mass of the ice is 30.7 g, the heat absorbed by the ice can be calculated as:

Q_ice = 30.7 g * 334 J/g = 10248.8 J

The heat lost by the water (Q_water) can be calculated using the equation:

Q_water = mass_water * heat_capacity_water * temperature_change_water

Given that the mass of the water is 0.189 kg, and the initial temperature of the water is 21.0°C, we need to find the temperature change of the water (temperature_change_water) to calculate the heat lost by the water.

To calculate the temperature change, we can use the equation:

temperature_change_water = final_temperature - initial_temperature

Since the final temperature is what we are trying to find, we will leave it as 'T' for now. Therefore,

temperature_change_water = T - 21.0°C

Now, we can rewrite the equation for the heat lost by the water as:

Q_water = 0.189 kg * heat_capacity_water * (T - 21.0°C)

According to the principle of conservation of energy, the heat absorbed by the ice is equal to the heat lost by the water. Therefore,

Q_ice = Q_water

Substituting the known values and rearranging the equation, we can solve for the final temperature of the water (T):

10248.8 J = 0.189 kg * heat_capacity_water * (T - 21.0°C)

Dividing both sides of the equation by 0.189 kg * heat_capacity_water, we get:

10248.8 J / (0.189 kg * heat_capacity_water) = T - 21.0°C

Now, we need to know the specific heat capacity of water. The specific heat capacity of water is approximately 4.186 J/g°C.

Substituting this value, we have:

10248.8 J / (0.189 kg * 4.186 J/g°C) = T - 21.0°C

Simplifying the equation:

T - 21.0°C = 1223.33

Adding 21.0°C to both sides, we get:

T = 1223.33 + 21.0°C

T ≈ 1244.33°C

Therefore, the final temperature of the water is approximately 1244.33°C.

To solve this problem, we need to use the concept of heat transfer and the specific heat capacity formula:

Q = m * c * ΔT

where:
Q = heat absorbed or released (in Joules)
m = mass (in kilograms)
c = specific heat capacity (in J/g°C or J/kg°C)
ΔT = change in temperature (in °C)

First, we need to calculate the heat absorbed by the ice:

Q_ice = m_ice * ΔH_fusion

where:
Q_ice = heat absorbed by the ice (in Joules)
m_ice = mass of the ice (in grams)
ΔH_fusion = heat of fusion (in J/g)

Given: m_ice = 30.7 g and ΔH_fusion = 0.334 kJ/g = 0.334 * 1000 J/g = 334 J/g

Q_ice = 30.7 g * 334 J/g
Q_ice = 10238.8 J

Next, we need to calculate the heat released by the water:

Q_water = m_water * c_water * ΔT_water

where:
Q_water = heat released by the water (in Joules)
m_water = mass of the water (in kilograms)
c_water = specific heat capacity of water (in J/g°C or J/kg°C)
ΔT_water = change in temperature of water (in °C)

Given: m_water = 0.189 kg and initial temperature = 21.0°C

To find the final temperature, we assume that the heat lost by the water is equal to the heat gained by the ice (Q_ice = Q_water).

Q_water = Q_ice
m_water * c_water * ΔT_water = 10238.8 J

Rearranging the equation, we can solve for ΔT_water:

ΔT_water = 10238.8 J / (m_water * c_water)

Given that c_water = 4.184 J/g°C, we need to convert the mass of water from kilograms to grams:

m_water = 0.189 kg * 1000 g/kg
m_water = 189 g

Substituting the values:

ΔT_water = 10238.8 J / (189 g * 4.184 J/g°C)
ΔT_water = 13.7419 °C

Therefore, the change in temperature of the water is 13.7419 °C.

To find the final temperature, we subtract the change in temperature from the initial temperature:

Final temperature = initial temperature - change in temperature
Final temperature = 21.0 °C - 13.7419 °C
Final temperature = 7.2581 °C

The final temperature of the water is 7.2581 °C.