Suppose 0.0150 kg of steam (at 100.00°C) is added to 0.150 kg of water (initially at 19.3°C.). The water is inside a copper cup of mass 54.1 g. The cup is inside a perfectly insulated calorimetry container that prevents heat flow with the outside environment. Find the final temperature (in °C) of the water after equilibrium is reached.
To find the final temperature of the water after equilibrium is reached, we can use the principle of conservation of energy.
First, we need to determine the initial energy of the steam and water before they reach equilibrium. We can use the equation:
Q = mcΔT
Where Q is the heat gained or lost, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
For the steam, we have:
Q₁ = m₁c₁ΔT₁
where:
m₁ = 0.0150 kg (mass of steam)
c₁ = 1.996 kJ/kgC (specific heat capacity of steam)
ΔT₁ = T₁ - 100.00°C (initial temperature of steam - final temperature)
For the water, we have:
Q₂ = m₂c₂ΔT₂
where:
m₂ = 0.150 kg (mass of water)
c₂ = 4.186 kJ/kg°C (specific heat capacity of water)
ΔT₂ = T₂ - 19.3°C (initial temperature of water - final temperature)
Next, we need to account for the heat exchanged between the steam and water, which is given by:
Q₃ = mcΔT
where:
m = mass of copper cup + water
c = specific heat capacity of copper
ΔT = final temperature - initial temperature
To find the final temperature, we set the total heat gained by the water equal to the total heat lost by the steam:
Q₁ + Q₂ = Q₃
Substituting the equations, rearranging, and solving for ΔT:
m₁c₁ΔT₁ + m₂c₂ΔT₂ = mcΔT
Plugging in the known values:
(0.0150 kg)(1.996 kJ/kg°C)(T₁ - 100.00°C) + (0.150 kg)(4.186 kJ/kg°C)(T₂ - 19.3°C) = (0.2041 kg)(0.385 kJ/kg°C)(T₂ - T₁)
Simplifying the equation:
1.996(T₁ - 100.00°C) + 0.6279(T₂ - 19.3°C) = 0.07879(T₂ - T₁)
Expanding and rearranging:
1.996T₁ - 199.40 + 0.6279T₂ - 12.01 = 0.07879T₂ - 0.07879T₁
Combining like terms:
1.996T₁ + 0.6279T₂ - 0.07879T₁ - 0.07879T₂ = 199.40 + 12.01
1.9172T₁ + 0.54911T₂ = 211.41
We also know that the total mass, m, is the sum of the mass of the copper cup and the water:
m = 0.2041 kg + 0.150 kg = 0.3541 kg
Next, we substitute this value into the equation for Q₃:
mcΔT = (0.3541 kg)(0.385 kJ/kg°C)(T₂ - T₁)
Simplifying:
0.137 kJ(T₂ - T₁) = 211.41
Dividing both sides by 0.137:
T₂ - T₁ = 1544.82
Finally, we arrange the equation so that it gives us the final temperature, T₂:
T₂ = T₁ + 1544.82
Now, we can solve for the final temperature (T₂) by assuming an initial temperature (T₁), substituting it into the equation, and calculating T₂. Repeat this process until you find the value of T₁ that satisfies the equation.