A 0.0400 kg ice cube at -30.0°C is placed in 0.400 kg of 35.0°C water in a very well insulated container. What is the final temperature?

To find the final temperature, we can use the principle of conservation of energy, specifically the principle of heat transfer.

The heat lost by the water is equal to the heat gained by the ice cube. The equation to calculate heat transfer is:

Q = m * c * ΔT

where Q is the heat transfer, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

For the water, the equation becomes:

Q_water = m_water * c_water * ΔT_water

For the ice cube, the equation becomes:

Q_ice = m_ice * c_ice * ΔT_ice

Since the container is very well insulated, the heat lost by the water is gained by the ice cube, so:

Q_water = Q_ice

Substituting the equation for heat transfer:

m_water * c_water * ΔT_water = m_ice * c_ice * ΔT_ice

Now, let's calculate the values:

m_water = 0.400 kg (mass of water)
c_water = 4186 J/kg°C (specific heat capacity of water)
ΔT_water = final temperature - initial temperature (unknown)

m_ice = 0.0400 kg (mass of ice)
c_ice = 2100 J/kg°C (specific heat capacity of ice)
ΔT_ice = final temperature - (-30.0°C) = final temperature + 30.0°C (unknown)

Since the final temperature is the same for both water and ice, we can simplify the equation to:

m_water * c_water * ΔT_water = m_ice * c_ice * ΔT_ice

(0.400 kg) * (4186 J/kg°C) * ΔT_water = (0.0400 kg) * (2100 J/kg°C) * (ΔT_ice + 30.0°C)

Simplifying further:

(0.400 kg * 4186 J/kg°C) * ΔT_water = (0.0400 kg * 2100 J/kg°C) * (ΔT_ice + 30.0°C)

Solving for ΔT_water:

ΔT_water = [(0.0400 kg * 2100 J/kg°C) * (ΔT_ice + 30.0°C)] / (0.400 kg * 4186 J/kg°C)

Now, we can find ΔT_water and then the final temperature by substituting the known values:

ΔT_water = [(0.0400 kg * 2100 J/kg°C) * (ΔT_ice + 30.0°C)] / (0.400 kg * 4186 J/kg°C)

ΔT_water = (168 J/°C * (ΔT_ice + 30.0°C)) / (836.4 J/°C)

Since we know that Q_water = Q_ice, we can set the heat transfer equations equal to each other:

m_water * c_water * ΔT_water = m_ice * c_ice * ΔT_ice

(0.400 kg) * (4186 J/kg°C) * ΔT_water = (0.0400 kg) * (2100 J/kg°C) * ΔT_ice

Now we can plug in the value we found for ΔT_water and solve for ΔT_ice:

(0.400 kg) * (4186 J/kg°C) * [(168 J/°C * (ΔT_ice + 30.0°C)) / (836.4 J/°C)] = (0.0400 kg) * (2100 J/kg°C) * ΔT_ice

After rearranging the equation and solving for ΔT_ice, we can find the final temperature by adding ΔT_ice to the initial temperature of the ice cube, which was -30.0°C.