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 \(1.0 \mathrm{~mol} \mathrm{Ag}\) 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

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·°C. c. The mass of the silver sample is 16250.0 g.

Step-by-step solution

Identify the given information and formula

The given information is: Specific heat capacity of silver (c) = 0.24 J/°C·g, Mass of silver (m) = 150.0 g, Initial temperature (T₁) = 273 K, Final temperature (T₂) = 298 K To calculate the energy required, use the formula: Energy (Q) = mc(T₂ - T₁)

Calculate the energy required

Substitute the given values into the formula: Q = (150.0 g) (0.24 J/°C·g) (298 K - 273 K) Q = (150.0 g) (0.24 J/°C·g) (25 K) 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 mol Ag by 1.0°C (called the molar heat capacity of silver).

Identify the given information and formula

The given information is: Specific heat capacity of silver (c) = 0.24 J/°C·g, Molar mass of silver (M) = 107.87 g/mol To calculate the molar heat capacity, use the formula: Molar heat capacity = c × M

Calculate the molar heat capacity

Substitute the given values into the formula: Molar heat capacity = (0.24 J/°C·g) (107.87 g/mol) Molar heat capacity = 25.89 J/mol·°C The molar heat capacity of silver is 25.89 J/mol·°C. 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.

Identify the given information and formula

The given information is: Specific heat capacity of silver (c) = 0.24 J/°C·g, Initial temperature (T₁) = 12.0°C, Final temperature (T₂) = 15.2°C, Energy (Q) = 1.25 kJ = 1250 J To calculate the mass of the silver sample, use the formula: Mass (m) = Q / (c × (T₂ - T₁))

Calculate the mass of the silver sample

Substitute the given values into the formula: m = 1250 J / [(0.24 J/°C·g)(15.2°C-12.0°C)] m = 1250 J / [(0.24 J/°C·g)(3.2°C)] m = 16250.0 g The mass of the silver sample is 16250.0 g.

Key concepts

Molar Heat Capacity

Molar heat capacity measures the amount of heat needed to raise the temperature of one mole of a substance by one degree Celsius. It's an important property for understanding how materials respond to heat. Silver has a molar heat capacity that can be calculated using its specific heat capacity and molar mass.
For silver, with a specific heat capacity of 0.24 J/°C·g and a molar mass of 107.87 g/mol, the molar heat capacity becomes:

• Molar Heat Capacity = Specific Heat Capacity \(c\) \( \times \) Molar Mass \(M\)
• Molar Heat Capacity = (0.24 J/°C·g) \( \times \) (107.87 g/mol)
• Molar Heat Capacity = 25.89 J/mol·°C

This means for each mole of silver, it takes 25.89 Joules to increase the temperature by one degree Celsius.

Energy Calculation

Energy calculations help determine the heat required to change the temperature of a substance. To calculate the energy required, we use the formula:

• Energy (Q) = mass \( \times \) specific heat capacity \( \times \) temperature change

In the example with silver, raising the temperature of 150 grams of silver from 273 K to 298 K involves:

• Q = (150.0 \(g\)) \( \times \) (0.24 \(J/°C\cdot g\)) \( \times \) (298 \(K\) - 273 \(K\))
• Q = 900 Joules

Thus, 900 J of energy is needed to make this temperature change.

Silver Properties

Silver, denoted by the symbol Ag, is a precious metal known for its high conductivity and thermal properties. The specific heat capacity is a thermodynamic property that indicates how efficiently a material can absorb heat.
Silver's specific heat capacity is relatively low at 0.24 J/°C·g, meaning it doesn't require a lot of energy to change its temperature. This characteristic makes it useful for various applications, including electronics and thermal conductivity uses.
Using its specific heat capacity, we can also determine other properties such as molar heat capacity or calculate the energy needed for temperate changes.

Mass Calculation

Accurate mass calculations of silver samples based on energy input can sound complex, but they follow a straightforward formula:

• Mass (m) = Energy (Q) / (Specific heat capacity \( \times \) Temperature change)

Given an energy input of 1.25 kJ (or 1250 J) to raise the temperature of silver from 12°C to 15.2°C, we can find the mass:

• m = 1250 J / (0.24 J/°C·g \( \times \) (15.2°C - 12.0°C))
• m = 1250 J / (0.24 J/°C·g \( \times \) 3.2°C)
• m = 16250.0 g

This calculation reveals that you have 16250 grams of silver, showcasing how mass is directly linked to energy changes and specific heat capacity.