Equal 0.400-kg mass of lead and tin at 60 degrees celsius are placed in 1kg of water at 20 degrees celsius.

a) what is the equilibrium temperature of the system?
b)if an alloy is half lead and half tin by mass, what specific heat would you anticipate for the alloy?
c) how many atoms of tin are in 0.400kg of tin and how many atoms of lead are in 0.400kg of lead?
d) divide the number of tin atoms by the number of lead atoms and compare this ratio with the specific heat of tin divided by the specific heat of lead. what conclusion can be drawn?

a) To find the equilibrium temperature of the system, we need to use the conservation of energy principle. The heat gained by water should be equal to the heat lost by the lead and tin:

Q_water = Q_lead + Q_tin

We know that Q = mcΔT, where m is the mass, c is the specific heat, and ΔT is the change in temperature.

For water, m_water = 1 kg, c_water = 4186 J/(kg·K), and ΔT_water = T_eq - 20 °C.

For lead, m_lead = 0.400 kg, c_lead = 128 J/(kg·K), and ΔT_lead = 60 °C - T_eq.

For tin, m_tin = 0.400 kg, c_tin = 227 J/(kg·K), and ΔT_tin = 60 °C - T_eq.

Now plug in the values and solve for T_eq:

(1 kg)(4186 J/(kg·K))(T_eq - 20 °C) = (0.400 kg)(128 J/(kg·K))(60 °C - T_eq) + (0.400 kg)(227 J/(kg·K))(60 °C - T_eq)

T_eq ≈ 30.64 °C

b) If an alloy is half lead and half tin by mass, the specific heat can be found using the weighted average of the specific heats:

c_alloy = (0.5)(c_lead) + (0.5)(c_tin) ≈ (0.5)(128 J/(kg·K)) + (0.5)(227 J/(kg·K)) ≈ 177.5 J/(kg·K)

c) To find the number of atoms in a mass, we can use the formula:

Number of atoms = (mass × Avogadro's number) / molar mass

For tin:

Number of tin atoms = (0.400 kg × 6.022 × 10^23 atoms/mol) / (118.71 g/mol × 1 kg/1000 g) ≈ 2.03 × 10^24 atoms

For lead:

Number of lead atoms = (0.400 kg × 6.022 × 10^23 atoms/mol) / (207.2 g/mol × 1 kg/1000 g) ≈ 1.16 × 10^24 atoms

d) Divide the number of tin atoms by the number of lead atoms:

(2.03 × 10^24 atoms) / (1.16 × 10^24 atoms) ≈ 1.75

Now divide the specific heat of tin by the specific heat of lead:

(227 J/(kg·K)) / (128 J/(kg·K)) ≈ 1.77

Both ratios are roughly equal. This implies that the number of atoms of each element in the mixture directly affects the specific heat of the alloy. In this case, it suggests that the specific heat of the alloy is influenced by both the specific heat and the relative proportion of each element.

a) To find the equilibrium temperature, we can use the principle of conservation of energy. The initial heat gained by the water should be equal to the total heat lost by the lead and tin alloys.

The heat gained by water (Qw) can be calculated using the formula:

Qw = m × c × ΔT

Where:
m = mass of water = 1 kg
c = specific heat capacity of water = 4.18 J/g°C (approximate value)
ΔT = change in temperature = Tf - Ti (final temperature - initial temperature)

Qw = 1 kg × 4.18 J/g°C × (Tf - 20°C)

The heat lost by the lead and tin alloys is the sum of the heat lost by each:

Ql = ml × cl × ΔT
Qt = mt × ct × ΔT

Where:
ml = mass of lead = 0.400 kg
cl = specific heat capacity of lead = 0.128 J/g°C (approximate value)
mt = mass of tin = 0.400 kg
ct = specific heat capacity of tin = 0.226 J/g°C (approximate value)

Substituting the given values and rearranging the equation, we have:

Qw = Ql + Qt
1 kg × 4.18 J/g°C × (Tf - 20°C) = 0.400 kg × 0.128 J/g°C × (Tf - 60°C) + 0.400 kg × 0.226 J/g°C × (Tf - 60°C)

Simplifying the equation further:

4.18 Tf - 83.6 = 0.0512 Tf - 2.048 + 0.0904 Tf - 3.616

Combining like terms:

4.18 Tf - 83.6 = 0.1416 Tf - 5.664

Subtracting 0.1416 Tf from both sides:

4.0384 Tf - 83.6 = -5.664

Adding 83.6 to both sides:

4.0384 Tf = 77.936

Dividing by 4.0384:

Tf ≈ 19.271°C

Therefore, the equilibrium temperature of the system is approximately 19.271°C.

b) Since the alloy is half lead and half tin by mass, we can expect the specific heat of the alloy (ca) to be the average of the specific heats of lead (cl) and tin (ct).

ca = (cl + ct) / 2
ca = (0.128 J/g°C + 0.226 J/g°C) / 2
ca ≈ 0.177 J/g°C

Therefore, we can anticipate a specific heat of approximately 0.177 J/g°C for the alloy.

c) To calculate the number of atoms in a given mass of a substance, we need to use the concept of molar mass and Avogadro's number.

The molar mass of tin (Sn) is 118.71 g/mol, and the molar mass of lead (Pb) is 207.2 g/mol.

For tin:
Number of moles of tin = mass of tin / molar mass of tin
Number of moles of tin = 0.400 kg × 1000 g/kg / 118.71 g/mol

Using Avogadro's number (6.022 × 10^23 atoms/mol), we can calculate the number of atoms of tin:

Number of atoms of tin = Number of moles of tin × Avogadro's number

For lead:
Number of moles of lead = mass of lead / molar mass of lead
Number of moles of lead = 0.400 kg × 1000 g/kg / 207.2 g/mol

Using Avogadro's number, we can calculate the number of atoms of lead:

Number of atoms of lead = Number of moles of lead × Avogadro's number

d) To compare the ratio of tin atoms to lead atoms with the ratio of specific heats of tin to lead, we need to calculate the ratios.

Ratio of tin atoms to lead atoms = Number of atoms of tin / Number of atoms of lead

Ratio of specific heats of tin to lead = Specific heat of tin / Specific heat of lead

By comparing these two ratios, we can draw a conclusion.

a) To find the equilibrium temperature of the system, we can use the principle of conservation of energy. The heat gained by one substance is equal to the heat lost by the other. We can calculate this using the formula:

Q_lead + Q_tin = Q_water

Where Q_lead and Q_tin are the heat absorbed by the lead and tin respectively, and Q_water is the heat released by the water.

To calculate the heat absorbed or released, we can use the specific heat capacity formula:

Q = m * c * ΔT

Where Q is the heat absorbed or released, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

Given:
Mass of lead (m_lead) = 0.400 kg
Mass of tin (m_tin) = 0.400 kg
Mass of water (m_water) = 1 kg
Initial temperature of lead and tin (T_lead, T_tin) = 60 degrees Celsius
Initial temperature of water (T_water) = 20 degrees Celsius

Using the formula, we can calculate the heat absorbed or released by each substance:

Q_lead = m_lead * c_lead * ΔT_lead
Q_tin = m_tin * c_tin * ΔT_tin
Q_water = m_water * c_water * ΔT_water

Since the equilibrium temperature is the same for all substances, ΔT_lead = ΔT_tin = ΔT_water = ΔT.

We can set up an equation using the principle of conservation of energy:

Q_lead + Q_tin = Q_water

m_lead * c_lead * ΔT + m_tin * c_tin * ΔT = m_water * c_water * ΔT

Substituting the given values, we have:

0.400 kg * c_lead * ΔT + 0.400 kg * c_tin * ΔT = 1 kg * c_water * ΔT

Now we can cancel ΔT from both sides:

0.400 kg * c_lead + 0.400 kg * c_tin = 1 kg * c_water

Rearranging the equation to solve for c_water:

c_water = (0.400 kg * c_lead + 0.400 kg * c_tin) / (1 kg)

This will give us the specific heat capacity of water.

b) If the alloy is half lead and half tin by mass, then the specific heat capacity of the alloy (c_alloy) can be estimated as the average of the specific heat capacities of lead (c_lead) and tin (c_tin).

c_alloy = (c_lead + c_tin) / 2

c) To find the number of atoms of tin and lead in the given mass, we need to use the concept of molar mass and Avogadro's number.

The molar mass of an element is the mass of one mole of atoms of that element. The molar mass of tin (M_tin) is approximately 118.71 g/mol, and the molar mass of lead (M_lead) is approximately 207.2 g/mol.

To calculate the number of moles (n) in a given mass (m), we can use the formula:

n = m / M

For tin:

m_tin = 0.400 kg = 400 g
n_tin = m_tin / M_tin

For lead:

m_lead = 0.400 kg = 400 g
n_lead = m_lead / M_lead

To convert the number of moles (n) to the number of atoms (N), we use Avogadro's number (N_A):

N = n * N_A

Where N_A is approximately 6.022 x 10^23 mol^-1.

For tin:

N_tin = n_tin * N_A

For lead:

N_lead = n_lead * N_A

d) To compare the ratio of tin atoms to lead atoms with the ratio of the specific heats, we divide the number of tin atoms by the number of lead atoms and compare it with the ratio of the specific heat of tin to the specific heat of lead.

Ratio of tin atoms to lead atoms:

(N_tin) / (N_lead) = (n_tin * N_A) / (n_lead * N_A)

Simplifying:

(N_tin) / (N_lead) = (n_tin) / (n_lead)

Ratio of specific heat of tin to specific heat of lead:

(c_tin) / (c_lead)

We can compare these two ratios to draw a conclusion.