A sample of steam with a mass of 0.552 g and at a temperature of 100 C condenses into an insulated container holding 4.45 g of water at 7.0 C.
Assuming that no heat is lost to the surroundings, what will be the final temperature of the mixture?
The sum of the heats gained is zero.
masssteam*Hvapor+masssteam*c*(Tf-100)+masswater(Tf-7)=0
solve for Tf.
17.7
To solve this problem, we can use the principle of conservation of energy. The heat lost by the steam will be equal to the heat gained by the water. We can calculate this using the equation:
Q (heat lost by steam) = Q (heat gained by water)
The heat lost by the steam can be calculated using the equation:
Q = mcΔT
where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Given:
Mass of steam (m1) = 0.552 g
Temperature of steam (T1) = 100 °C
Mass of water (m2) = 4.45 g
Initial temperature of water (T2) = 7.0 °C
First, we need to calculate the heat lost by the steam:
Q1 = m1 * c1 * ΔT1
The specific heat capacity of steam (c1) is 2.03 J/g°C.
Next, we can calculate the heat gained by the water:
Q2 = m2 * c2 * ΔT2
The specific heat capacity of water (c2) is 4.18 J/g°C.
Since the process is insulated, the heat lost by the steam is equal to the heat gained by the water:
Q1 = Q2
Now, we can solve for the final temperature of the mixture (T_f) using the equation:
Q1 = m2 * c2 * (T_f - T2)
Rearranging the equation:
T_f = (Q1 / (m2 * c2)) + T2
Let's calculate the values step-by-step:
Step 1: Calculating the heat lost by the steam (Q1):
Q1 = m1 * c1 * ΔT1
Q1 = 0.552 g * 2.03 J/g°C * (100 °C - Tf)
Step 2: Calculating the heat gained by the water (Q2):
Q2 = m2 * c2 * ΔT2
Q2 = 4.45 g * 4.18 J/g°C * (Tf - 7.0 °C)
Step 3: Setting the heat lost by the steam equal to the heat gained by the water:
Q1 = Q2
Step 4: Solve for Tf:
0.552 g * 2.03 J/g°C * (100 °C - Tf) = 4.45 g * 4.18 J/g°C * (Tf - 7.0 °C)
Simplify the equation:
1.1156 g °C * (100 °C - Tf) = 18.581 g °C * (Tf - 7.0 °C)
Solve for Tf:
111.56 °C - 1.1156 °C Tf = 18.581 °C Tf - 130.067 °C
Combine similar terms:
19.6966 °C Tf = 241.627 °C
Solve for Tf:
Tf = (241.627 °C) / (19.6966 °C)
Tf ≈ 12.27 °C
Therefore, the final temperature of the mixture will be approximately 12.27 °C.
To find the final temperature of the mixture, we need to apply the principle of conservation of energy, which states that the total energy of a closed system remains constant.
We can use the equation:
Q_lost = Q_gained
where Q_lost is the heat lost by the steam and Q_gained is the heat gained by the water.
First, we need to calculate the heat lost by the steam. This can be done using the formula:
Q_lost = m * c * ΔT
where m is the mass of the steam, c is the specific heat capacity of steam, and ΔT is the change in temperature.
The specific heat capacity of steam is generally considered to be equal to the specific heat capacity of water, which is approximately 4.18 J/g·C.
ΔT for the steam can be calculated as:
ΔT = final temperature - initial temperature
ΔT = 100 C - final temperature
Substituting the given values:
Q_lost = 0.552 g * 4.18 J/g·C * (100 C - final temperature)
Next, we need to calculate the heat gained by the water. This can also be calculated using the formula:
Q_gained = m * c * ΔT
where m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
ΔT for the water can be calculated as:
ΔT = final temperature - initial temperature
ΔT = final temperature - 7.0 C
Substituting the given values:
Q_gained = 4.45 g * 4.18 J/g·C * (final temperature - 7.0 C)
Since the system is insulated and assuming no heat is lost to the surroundings, Q_lost = Q_gained. So we can set up the equation:
0.552 g * 4.18 J/g·C * (100 C - final temperature) = 4.45 g * 4.18 J/g·C * (final temperature - 7.0 C)
Now, we can solve this equation for the final temperature.