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))

To solve these problems, we'll need to apply the formulas related to thermal expansion for the first question, and use the principles of calorimetry for the second question.

1. To find the temperature to which the rivet must be cooled in order to fit in the hole, we can use the formula for thermal expansion:

ΔL = α * L * ΔT

Where:
ΔL is the change in length or diameter,
α is the coefficient of linear expansion,
L is the initial length or diameter, and
ΔT is the change in temperature.

In this case, we want to find the change in temperature (ΔT). We can rearrange the formula to solve for ΔT:

ΔT = ΔL / (α * L)

Given that the diameter of the rivet (initial length) is 1.871 cm, the diameter of the hole (final length) is 1.869 cm, and the coefficient of linear expansion (α) for steel is approximately 11 x 10^(-6) °C^(-1), we can substitute these values into the formula to find ΔT:

ΔT = (1.871 cm - 1.869 cm) / (11 x 10^(-6) °C^(-1) * 1.871 cm)

Calculating this expression will give us the change in temperature.

2. For this question, we'll use the principles of calorimetry. The total heat gained by the water and the cup should be equal to the heat lost by the alloy:

m_water * C_water * ΔT_water + m_cup * C_cup * ΔT_cup = m_alloy * C_alloy * ΔT_alloy

Where:
m is the mass of the substance,
C is the specific heat capacity of the substance, and
ΔT is the change in temperature.

In this case, we want to find the specific heat of the alloy (C_alloy). Rearranging the formula, we can solve for C_alloy:

C_alloy = (m_water * C_water * ΔT_water + m_cup * C_cup * ΔT_cup) / (m_alloy * ΔT_alloy)

Given the values:
m_alloy = 0.150 kg
ΔT_water = 30.5 °C - 10 °C = 20.5 °C
m_water = 400 g = 0.400 kg
C_water = 4186 J/(kg·°C)
m_cup = 200 g = 0.200 kg
C_cup = 900 J/(kg·°C)
ΔT_cup = 30.5 °C - 10 °C = 20.5 °C

Substituting these values into the formula will give us the specific heat of the alloy.

I will be happy to critique your thinking. I am not certain with what you are having difficulty.