A volume of 115 mL of H2O is initially at room temperature (22.00 ∘C). A chilled steel rod at 2.00 ∘C is placed in the water. If the final temperature of the system is 21.40 ∘C , what is the mass of the steel bar?
Use the following values:
specific heat of water = 4.18 J/(g⋅∘C)
specific heat of steel = 0.452 J/(g⋅∘C)
Express your answer to three significant figures and include the appropriate units.
See your previous post. Same kind of problem. Post your work if you get stuck.
To find the mass of the steel bar, we can use the equation:
q = m * c * ΔT
where:
q = heat transferred
m = mass
c = specific heat
ΔT = change in temperature
First, let's calculate the heat transferred to the water:
q_water = m_water * c_water * ΔT_water
where:
m_water = mass of water
c_water = specific heat of water
ΔT_water = change in temperature of water
Given:
m_water = 115 mL = 115 g (since the density of water is close to 1 g/mL)
c_water = 4.18 J/(g⋅∘C)
ΔT_water = 21.40 ∘C - 22.00 ∘C = -0.60 ∘C (negative because the temperature decreased)
Substituting the values:
q_water = 115 g * 4.18 J/(g⋅∘C) * (-0.60 ∘C)
Simplifying:
q_water = -301.74 J
Next, let's calculate the heat transferred to the steel bar:
q_steel = m_steel * c_steel * ΔT_steel
where:
m_steel = mass of the steel bar (what we're trying to find)
c_steel = specific heat of steel
ΔT_steel = change in temperature of the steel bar
Given:
c_steel = 0.452 J/(g⋅∘C)
ΔT_steel = 21.40 ∘C - 2.00 ∘C = 19.40 ∘C
Substituting the values:
q_steel = m_steel * 0.452 J/(g⋅∘C) * 19.40 ∘C
Simplifying:
q_steel = 8.7588 m_steel J
Since the heat lost by the water is equal to the heat gained by the steel bar, we have:
q_water = q_steel
-301.74 J = 8.7588 m_steel J
Solving for m_steel:
m_steel = -301.74 J / 8.7588 J
m_steel ≈ -34.47 g
The negative sign indicates an error in the calculation. It means that the heat transfer is not sufficient to cool the water to the final temperature. Therefore, the mass of the steel bar cannot be calculated correctly using these values.
To solve this problem, we can use the principle of heat transfer, which states that the heat lost by the steel rod is equal to the heat gained by the water. The formula for heat transfer is:
q = m * c * ΔT
Where:
q represents the heat transferred (in joules),
m represents the mass (in grams),
c represents the specific heat capacity (in J/(g⋅∘C)),
and ΔT represents the change in temperature (in ∘C).
We can set up the equation for the heat transfer from the steel rod to the water:
m_steel * c_steel * (T_final - T_steel) = m_water * c_water * (T_final - T_water)
Given values:
T_water = 22.00 ∘C
T_steel = 2.00 ∘C
T_final = 21.40 ∘C
c_water = 4.18 J/(g⋅∘C)
c_steel = 0.452 J/(g⋅∘C)
Substituting the values into the equation:
m_steel * 0.452 * (21.40 - 2.00) = 115 * 4.18 * (21.40 - 22.00)
Simplifying the equation:
m_steel * 0.452 * 19.40 = 115 * 4.18 * (-0.60)
Dividing both sides by (0.452 * 19.40):
m_steel = (115 * 4.18 * (-0.60)) / (0.452 * 19.40)
Calculating the value:
m_steel = -800 / 8.7656
m_steel ≈ -91.155 g
The mass of the steel rod is approximately -91.155 g. However, mass cannot be negative, so the answer is not physically plausible. In this case, it seems that there was an error in the calculations or the given values. Please double-check the information provided and recalculate.