Problem 50 - Chemistry An Atoms First Approach

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 \(150.0 \mathrm{g}\) Ag from \(273 \mathrm{K}\) to \(298 \mathrm{K}\). b. Calculate the energy required to raise the temperature of 1.0 mole of 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 silver.

Step-by-Step Solution

Answer

a. The energy required to raise the temperature of \(150.0 \, g\) Ag from \(273 \, K\) to \(298 \, K\) is \(900 \, J\). b. The molar heat capacity of silver is \(25.89 \, J/mol \cdot ^{o}C\). c. The mass of the sample silver is \(21708.3 \, g\).

Step 1: Part a: Calculate the energy required for a 150 g silver sample

The given data are: - mass (m): \(150.0 \, g\) - specific heat capacity (c): \(0.24 \, J/g \cdot ^{o}C\) - temperature change (\(\Delta T\)): \(298 \, K - 273 \, K = 25 \, ^{o}C\) Now, we will use the formula \(q = mc\Delta T\). \(q = (150.0 \, g) \times (0.24 \, J/g \cdot ^{o}C) \times (25 \, ^{o}C) = 900 \, J\) Therefore, it takes 900 Joules of energy to raise the temperature of \(150.0 \, g\) Ag from \(273 \, K\) to \(298 \, K\).

Step 2: Part b: Calculate molar heat capacity of silver

First, we need to find the molar mass of Silver (Ag): - Given, 1 mole Ag has a molar mass of \(107.87 \, g\). Now, we need to find the energy required to raise the temperature of 1 mole of Ag by \(1.0 \, ^{o}C\). We use the same formula, \(q = mc\Delta T\), but with the mass of 1 mole Ag. m = \(107.87 \, g\) \(\Delta T = 1 \, ^{o}C\) \(q_{1 mole} = (107.87 \, g) \times (0.24 \, J/g \cdot ^{o}C) \times (1 \, ^{o}C) = 25.89 \, J\) Therefore, the molar heat capacity of silver is \(25.89 \, J/mol \cdot ^{o}C\).

Step 3: Part c: Calculate the mass of the silver sample

The given data are: - energy (q): \(1.25 \, kJ = 1250 \, J\) - temperature change (\(\Delta T\)): \(15.2 \, ^{o}C - 12.0 \, ^{o}C = 3.2 \, ^{o}C\) - specific heat capacity (c): \(0.24 \, J/g \cdot ^{o}C\) We will again use the formula \(q = mc\Delta T\), but this time, we will need to solve for mass (m). \(1250 \, J = m \times (0.24 \, J/g \cdot ^{o}C) \times (3.2 \, ^{o}C)\) Now, we will isolate the mass (m) variable by dividing both sides of the equation by \(0.24 \, J/g \cdot ^{o}C\) and \(3.2\, ^{o}C\). \(m = \frac{1250 \, J}{(0.24 \, J/g \cdot ^{o}C) \times (3.2 \, ^{o}C)} = 21708.3 \, g\) Therefore, the mass of the sample silver is \(21708.3 \, g\).

Key Concepts

Molar Heat Capacity

Molar heat capacity is an important concept in chemistry as it provides insight into the amount of heat energy needed to increase the temperature of one mole of a substance by one degree Celsius. For silver, given its specific heat capacity is 0.24 J/g°C and its molar mass is 107.87 g/mol, we can calculate the molar heat capacity using the formula:

• q represents the heat energy absorbed or released.
• c is the specific heat capacity.
• ΔT is the temperature change.
• m is the mass in grams.

For one mole of silver, the mass (m) is 107.87 g. Hence, to find the molar heat capacity (C_m) in J/mol°C, you use:\[ C_m = c \times m = (0.24 \text{ J/g°C}) \times (107.87 \text{ g}) = 25.89 \text{ J/mol°C} \]This means that 25.89 Joules of energy are required to raise one mole of silver by 1°C.

Energy Calculation

Energy calculation in heating processes involves determining how much energy is needed to change a substance's temperature. The formula used is: \[ q = mcΔT \]This allows us to calculate the thermal energy (q) needed. Let's review what each part of the formula means:

• q: Energy in Joules.
• m: Mass of the substance in grams.
• c: Specific heat capacity, which is a constant that represents how much energy is needed to raise 1 gram of a substance by 1°C.
• ΔT: The change in temperature.

For instance, if you have 150 g of silver and you want to raise its temperature from 273 K to 298 K, you first convert the temperature change in Kelvin to Celsius (because specific heat capacity is per °C). Thus: \[ ΔT = 298 \text{ K} - 273 \text{ K} = 25 °\text{C} \]Now, insert the known values into the formula:\[ q = (150 \text{ g}) \times (0.24 \text{ J/g°C}) \times (25 °\text{C}) = 900 \text{ J} \]Therefore, it requires 900 Joules to heat the silver as described.

Temperature Change

Temperature change (ΔT) is a key component in calculations related to heat energy transfer in materials. It tells us how much the temperature has shifted between the start and end of a process. Knowing the change aids in calculating the energy required to achieve such a shift.

• You compute ΔT by subtracting the initial temperature from the final temperature.
• In problems involving specific heat, ΔT must be in degrees Celsius, as this aligns with the units of the specific heat capacity.

Let's take an example from the exercise: to find the mass of silver from the energy change, given that 1250 J of energy changes the temperature from 12°C to 15.2°C, we calculate \[ ΔT = 15.2 °\text{C} - 12.0 °\text{C} = 3.2 °\text{C} \]Now you can find the mass m by rearranging the energy formula and solving for m:\[ 1250 \text{ J} = m \times (0.24 \text{ J/g°C}) \times (3.2 °\text{C}) \]Solving for m gives: \[ m = \frac{1250 \text{ J}}{(0.24 \text{ J/g°C}) \times (3.2 °\text{C})} = 21708.3 \text{ g} \]This comprehensive understanding of ΔT facilitates accurate energy calculations.