Question

The specific heat capacity of silver is 0.24 \(\mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\) a. Calculate the energy required to raise the temperature of 150.0 g Ag from 273 \(\mathrm{K}\) to 298 \(\mathrm{K}\) . b. Calculate the energy required to raise the temperature of 1.0 mole of \(\mathrm{Ag}\) by \(1.0^{\circ} \mathrm{C}\) (called the molar heat capacity of silver). c. It takes 1.25 \(\mathrm{kJ}\) of energy to heat a sample of pure silver from \(12.0^{\circ} \mathrm{C}\) to \(15.2^{\circ} \mathrm{C}\) . Calculate the mass of the sample of silver.

Short answer

The energy required to raise the temperature of 150.0 g Ag from 273 K to 298 K is 900 J, the energy required to raise the temperature of 1.0 mole of Ag by 1.0°C is 25.89 J, and the mass of the sample of silver that takes 1.25 kJ of energy to heat from 12.0°C to 15.2°C is approximately 1625 g.

Step-by-step solution

a. Calculate the energy required to raise the temperature of 150.0 g Ag from 273 K to 298 K.

First, we need to convert the temperature from Kelvins to Celsius: 273 K = 0°C and 298 K = 25°C. Now use the formula Q = mcΔT: Mass (m) = 150.0 g Specific heat capacity (c) = 0.24 J/(g°C) Change in temperature (ΔT) = 25°C - 0°C = 25°C Q = (150.0 g) * (0.24 J/(g°C)) * (25°C) Q = 900 J The energy required to raise the temperature of 150.0 g Ag from 273 K to 298 K is 900 J.

b. Calculate the energy required to raise the temperature of 1.0 mole of Ag by 1.0°C.

First, we have to calculate the mass of 1.0 mole of Ag: Molar mass of Ag = 107.87 g/mol Mass (m) = (1.0 mol) * (107.87 g/mol) = 107.87 g Now, use the formula Q = mcΔT with ΔT = 1.0°C: Q = (107.87 g) * (0.24 J/(g°C)) * (1.0°C) Q = 25.89 J The energy required to raise the temperature of 1.0 mole of Ag by 1.0°C is 25.89 J.

c. It takes 1.25 kJ of energy to heat a sample of pure silver from 12.0°C to 15.2°C. Calculate the mass of the sample of silver.

First, we need to convert the energy from kJ to J: 1.25 kJ * 1000 = 1250 J Now we have the energy (Q), the specific heat capacity (c), and the change in temperature (ΔT = 15.2°C - 12.0°C). We can rewrite the formula Q = mcΔT to solve for mass (m): m = Q / (c * ΔT) m = (1250 J) / (0.24 J/(g°C) * (15.2°C - 12.0°C)) m = 1250 J / (0.24 J/(g°C) * 3.2°C) m ≈ 1625 g The mass of the sample of silver is approximately 1625 g.

Key concepts

Energy Calculation

Energy calculation involves determining the total amount of energy needed to cause a specific change in temperature for a substance. In our exercise, the goal is to calculate how much energy is needed to increase the temperature of a given mass of silver. To perform this calculation, we use the formula:
\[ Q = mc\Delta T \]where:

• \( Q \) is the energy transfer in joules (J).
• \( m \) is the mass of the substance in grams (g).
• \( c \) is the specific heat capacity in J/(g°C), which is the amount of energy required to heat 1 gram of a substance by 1°C.
• \( \Delta T \) is the change in temperature in degrees Celsius (°C).

For example, raising the temperature of 150.0 grams of silver from 273 K to 298 K means finding \( \Delta T \) in °C first.
This is done by converting the absolute temperature in kelvins to Celsius, resulting in a \( \Delta T \) of 25°C.
By substituting the values into the equation, we compute the energy \( Q \) needed, which is found to be 900 J. This demonstrates that energy calculation is crucial in thermodynamics to quantify the heat transfer.

Molar Heat Capacity

Molar heat capacity represents the amount of energy required to raise the temperature of 1 mole of a substance by 1 degree Celsius. It provides insights into the thermal properties of the substance on a molecular scale.
To calculate the molar heat capacity of silver, given its specific heat capacity, we also need to know the molar mass of silver, which is approximately 107.87 g/mol.
To find how much energy is needed to raise the temperature of 1 mole of silver by 1°C, the formula \( Q = mc\Delta T \) is adapted:

• \( m = 107.87 \) g for 1 mole of silver
• \( c = 0.24 \) J/(g°C)
• \( \Delta T = 1.0°C \)

Substituting in the values, we get:\[Q = (107.87 \, \text{g}) \times (0.24 \, \text{J/(g°C)}) \times (1.0 \, \text{°C}) = 25.89 \, \text{J}\]The molar heat capacity gives a specific focus on how substances of different molar mass and same temperature difference require differing amounts of energy, indicating the varying heat storage capacities at the molecular level.

Mass Calculation

Finding the mass of a sample given its energy intake and temperature change is an inverse application of the heat equation. In this exercise, we are provided with 1.25 kJ of energy (which converts to 1250 J) needed to increase the temperature of a sample of silver from 12.0°C to 15.2°C.
Given the specific heat capacity and temperature change, the mass \( m \) can be found by rearranging the formula for energy:\[ m = \frac{Q}{c\Delta T} \]where:

• \( Q = 1250 \) J
• \( c = 0.24 \) J/(g°C)
• \( \Delta T = 15.2°C - 12.0°C = 3.2°C \)

Inserting the known values, the formula becomes:\[ m = \frac{1250 \, \text{J}}{0.24 \, \text{J/(g°C)} \cdot 3.2 \, \text{°C}}\]Solving this gives \( m \approx 1625 \) grams.
This calculation helps determine the physical quantities involved when specific heat and temperature shifts are known, reinforcing understanding of practical thermochemistry applications.