Problem 93 - Chemistry The Molecular Science

Question

Calculate the quantity of energy, in joules, required to raise the temperature of \(454 \mathrm{~g}\) tin from room temperature, \(25.0^{\circ} \mathrm{C}\), to its melting point, \(231.9{ }^{\circ} \mathrm{C},\) and then melt the tin at that temperature. (The specific heat capacity of tin is \(0.227 \mathrm{~J} \mathrm{~g}^{-1} \mathrm{~K}^{-1}\), and the enthalpy of fusion of this metal is \(59.2 \mathrm{~J} / \mathrm{g} .\) )

Step-by-Step Solution

Answer

The total energy required is approximately 48185 J.

Step 1: Understand the Problem

We need to raise the temperature of tin from 25.0°C to its melting point, 231.9°C, and then melt it. Two separate calculations are needed: one for heating and one for melting.

Step 2: Calculate Energy Required to Heat Tin

To find the energy required to raise the temperature, use the formula: \[ q = m \times c \times \Delta T \]where \( q \) is the energy in joules, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature in Kelvin.Given:- \( m = 454 \) g- \( c = 0.227 \) J/g/K- \( \Delta T = 231.9 - 25.0 = 206.9 \) KSubstitute the values:\[ q = 454 \times 0.227 \times 206.9 = 21348.49 \text{ J} \]

Step 3: Calculate Energy Required to Melt Tin

To find the energy required to melt the tin at its melting point, use the formula: \[ q = m \times \Delta H_f \]where \( q \) is the energy, \( m \) is the mass, and \( \Delta H_f \) is the enthalpy of fusion.Given:- \( m = 454 \) g- \( \Delta H_f = 59.2 \) J/gSubstitute the values:\[ q = 454 \times 59.2 = 26836.8 \text{ J} \]

Step 4: Calculate Total Energy Required

The total energy required is the sum of the energy needed to heat the tin and the energy needed to melt it:\[ q_{total} = 21348.49 + 26836.8 = 48185.29 \text{ J} \]

Key Concepts

Specific Heat Capacity

When you want to heat a substance, one of the first things you need to know is its specific heat capacity. This is a measure that tells us how much energy needs to be added to raise the temperature of one gram of the substance by one degree Kelvin (or Celsius). The formula to calculate the energy required to heat a substance is:
\[ q = m \times c \times \Delta T \]where:

• \( q \) is the energy in joules,
• \( m \) is the mass in grams,
• \( c \) is the specific heat capacity,
• \( \Delta T \) is the change in temperature in Kelvin.

In our exercise, tin's specific heat capacity is given as \(0.227 \text{ J/g/K}\). To find the energy needed, multiply the mass of the tin by its specific heat capacity and the temperature change (\(206.9 \text{ K}\)). This tells us how much energy is necessary to heat tin until it reaches its melting point.

Enthalpy of Fusion

Once a substance reaches its melting point, another type of energy calculation comes into play. This is where enthalpy of fusion enters the picture.
Enthalpy of fusion is the amount of energy required to change a substance from solid to liquid at its melting point. The formula for this process is:
\[ q = m \times \Delta H_f \]where:

• \( q \) is the energy in joules,
• \( m \) is the mass,
• \( \Delta H_f \) is the enthalpy of fusion.

For tin, the enthalpy of fusion is \(59.2 \text{ J/g}\). To find the energy required to melt 454 grams of tin, you multiply the mass by the enthalpy of fusion. This calculation tells us how much energy is needed to convert all of the tin at its melting point from a solid state to a liquid state.

Energy Calculations

Energy calculations are crucial when examining thermal processes, as they help determine how much energy is necessary to complete a reaction or phase change.
In scenarios like the transition of tin from room temperature to liquid form, you start with two main steps. First, calculate the energy needed to raise the temperature to the melting point using specific heat capacity. Then, compute the energy necessary for melting using the enthalpy of fusion.
In our example:

• The energy to heat tin from \(25^{\circ} \text{C}\) to \(231.9^{\circ} \text{C}\) is \(21348.49 \text{ J}\).
• The energy to melt the tin at that temperature is \(26836.8 \text{ J}\).
• Adding both gives a total energy of \(48185.29 \text{ J}\).

These calculations reveal not just the individual requirements for each phase of heating and melting but also provide the total energy expenditure needed throughout the entire process.