1. to make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. a steel rivet 1.871 cm in diameter is to be placed in a hole 1.869 cm in diameter at 20 degree celcius. to what temperature must the rivet be cooled if it is to fit in the hole?
2. you wish to determine the specific heat of a new metal alloy. A 0.150 kg sample of the alloy is heated to 540 degree C. it is then quickly placed in 400 g of water at 10 degree C which is contained in a 200 g aluminum calorimeter cup. the final temp. of the system is 30.5 degree C. calculate the specific heat of the alloy. (Cwater = 4186 J/kgoC);Ccup=900 J/kgoC))
for number 1, is the answer negative?
For number 2, how do we solve it without the Calloy?
1. To determine the temperature to which the rivet must be cooled in order to fit in the hole, we can use the principle of thermal expansion. The basic idea is that objects expand when heated and contract when cooled. The amount of expansion or contraction can be calculated using the coefficient of linear expansion.
The formula for linear expansion is: ΔL = α * L * ΔT
Where:
ΔL is the change in length,
α is the coefficient of linear expansion,
L is the original length, and
ΔT is the change in temperature.
In this case, the rivet diameter will change due to the change in temperature. The change in diameter (Δd) can be calculated using the formula: Δd = 2 * α * d * ΔT
We need to find the change in temperature (ΔT) that will make the rivet fit into the hole. To do this, we can rearrange the formula to solve for ΔT:
ΔT = (Δd / (2 * α * d))
Now, let's plug in the values:
Given diameter of the rivet, d = 1.871 cm = 0.01871 m
Given diameter of the hole, d_hole = 1.869 cm = 0.01869 m
Coefficient of linear expansion for steel, α = 12 x 10^-6 / °C (approximately)
Δd = d - d_hole = 0.01871 m - 0.01869 m = 2 x 10^-5 m
Substituting these values into the formula:
ΔT = (2 x 10^-5) / (2 * 12 x 10^-6 * 0.01871) ≈ 4.23 °C
Therefore, the rivet must be cooled by approximately 4.23 °C to fit into the hole.
2. In this case, we can use the principle of conservation of energy to solve for the specific heat of the alloy. The heat gained by the water and the calorimeter cup will be equal to the heat lost by the alloy.
The formula for heat transfer is: Q = m * C * ΔT
Where:
Q is the heat transferred,
m is the mass of the substance,
C is the specific heat capacity, and
ΔT is the change in temperature.
We can start by calculating the heat gained by the water and the calorimeter cup.
Heat gained by Water: Q_water = m_water * C_water * ΔT_water
Given:
Mass of water, m_water = 400 g = 0.4 kg
Specific heat capacity of water, C_water = 4186 J/kg°C
Change in temperature for water, ΔT_water = 30.5 °C - 10 °C = 20.5 °C
Substituting these values into the formula:
Q_water = 0.4 kg * 4186 J/kg°C * 20.5 °C = 34168 J
Heat gained by Calorimeter Cup: Q_cup = m_cup * C_cup * ΔT_cup
Given:
Mass of calorimeter cup, m_cup = 200 g = 0.2 kg
Specific heat capacity of aluminum (cup), C_cup = 900 J/kg°C
Change in temperature for cup, ΔT_cup = 30.5 °C - 10 °C = 20.5 °C
Substituting these values into the formula:
Q_cup = 0.2 kg * 900 J/kg°C * 20.5 °C = 3690 J
The total heat gained by the water and the calorimeter cup is:
Q_total = Q_water + Q_cup = 34168 J + 3690 J = 37858 J
Since the heat gained by the water and the calorimeter cup is equal to the heat lost by the alloy, we can set up the equation:
Q_total = m_alloy * C_alloy * ΔT
Rearranging the equation:
C_alloy = Q_total / (m_alloy * ΔT)
Given:
Mass of alloy, m_alloy = 0.150 kg
Change in temperature for the system, ΔT = 30.5 °C - 540 °C = - 509.5 °C
Substituting these values into the formula:
C_alloy = 37858 J / (0.150 kg * -509.5 °C) ≈ -5.03 J/kg°C
Therefore, the specific heat of the alloy is approximately -5.03 J/kg°C.
Note: The negative value for the specific heat of the alloy indicates that it releases heat when its temperature decreases.
1, Yes
2. You are looking for Calloy, that is your unknown.