Problem 53 - Chemistry The Central Science In Si Units

Question

The specific heat of octane, \(\mathrm{C}_{8} \mathrm{H}_{18}(I),\) is \(2.22 \mathrm{~J} / \mathrm{g}-\mathrm{K} .(\mathrm{a}) \mathrm{How}\) many J of heat are needed to raise the temperature of \(80.0 \mathrm{~g}\) of octane from 10.0 to \(25.0^{\circ} \mathrm{C} ?\) (b) Which will require more heat, increasing the temperature of \(1 \mathrm{~mol}\) of \(\mathrm{C}_{8} \mathrm{H}_{18}(I)\) by a certain amount or increasing the temperature of \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(I)\) by the same amount?

Step-by-Step Solution

Answer

(a) 2664 J of heat is needed for octane. (b) More heat is required for 1 mol of octane than for water.

Step 1: Understand the formula for heat energy

To find the amount of heat required to change the temperature of a substance, we use the formula: \[ q = m \cdot c \cdot \Delta T \] where \( q \) is the heat energy (in Joules), \( m \) is the mass (in grams), \( c \) is the specific heat capacity (in J/g-K), and \( \Delta T \) is the change in temperature (in Kelvin or Celsius).

Step 2: Calculate the change in temperature

The change in temperature \( \Delta T \) is calculated by subtracting the initial temperature from the final temperature: \[ \Delta T = \text{final temperature} - \text{initial temperature} = 25.0^{\circ}C - 10.0^{\circ}C = 15.0^{\circ}C \]

Step 3: Insert values to find heat required for octane

Using the formula from Step 1 and the values given, \( m = 80.0 \) g, \( c = 2.22 \text{ J/g-K} \), and \( \Delta T = 15.0^{\circ}C \), we can calculate: \[ q = 80.0 \cdot 2.22 \cdot 15.0 = 2664 \text{ J} \] Thus, 2664 Joules of heat are required to raise the temperature of 80.0 g of octane from 10.0°C to 25.0°C.

Step 4: Understanding heat requirement for 1 mol substances

The heat required to increase the temperature of 1 mol of a substance depends on the molar mass and specific heat. Molar mass of \( \mathrm{C}_{8} \mathrm{H}_{18} \) is approximately 114 g/mol. For \( \mathrm{H}_{2} \mathrm{O} \), molar mass is approximately 18 g/mol with specific heat approximately 4.18 J/g-K.

Step 5: Compare heat for 1 mol octane and water

For 1 mol of \( \mathrm{C}_{8} \mathrm{H}_{18} \), heat is: \[ q = 114 \cdot 2.22 \cdot \Delta T \]. For 1 mol of \( \mathrm{H}_{2} \mathrm{O} \), heat is: \[ q = 18 \cdot 4.18 \cdot \Delta T \]. Calculate: \( 114 \cdot 2.22 = 253.08 \) and \( 18 \cdot 4.18 = 75.24 \). \( 253.08 > 75.24 \), so more heat is needed for octane.

Key Concepts

Heat Energy Calculation

To calculate the heat energy required to change the temperature of a substance, we turn to the formula:

• \( q = m \cdot c \cdot \Delta T \)

Here, \( q \) represents the heat energy in Joules, \( m \) is the mass of the substance in grams, \( c \) stands for the specific heat capacity in Joules per gram per Kelvin (J/g-K), and \( \Delta T \) is the temperature change in either Celsius or Kelvin. Both units of temperature lead to the same result as the change in value, not the absolute value, is what contributes. It's always important to ensure that the units are consistent before you plug values into the formula.
For instance, in the case of octane, if you have 80 g of the substance and you are raising the temperature from 10°C to 25°C, the change in temperature \( \Delta T \) would be 15°C. Substituting these values into the formula gives:

• \( q = 80.0 \times 2.22 \times 15.0 = 2664 \text{ J} \)

So, 2664 Joules of heat is needed. Remember, the choice of specific heat in units of J/g-K is a measure of how much heat a gram of substance needs to increase the temperature by 1 K or 1°C. This shows how efficiently the substance uses energy for temperature change.

Temperature Change

Temperature change, denoted by \( \Delta T \), is a crucial component in calculating heat energy. It is computed by subtracting the initial temperature from the final temperature:

• \( \Delta T = \text{final temperature} - \text{initial temperature} \)

This implies that if you start with a temperature of 10°C and increase it to 25°C, your temperature change \( \Delta T \) is 15°C. Temperature change tells us how much "push" or energy is required to reach the new temperature. While measuring \( \Delta T \), it is important to focus on the actual change between numbers, rather than the units themselves, whether they are in Celsius or Kelvin. While Celsius is usually used in everyday context, Kelvin is commonly used in scientific measurements but the delta or change remains the same. This ensures calculations are consistent when utilizing the specific heat capacity formula.

Molar Mass Comparison

When comparing the heat requirement of different substances, molar mass becomes invaluable. Different substances have different masses in moles, which influences the heat needed to achieve a temperature change. The concept lies in their respective formulas:

• For a mole of a substance, \( q = \text{Molar Mass} \times \text{Specific Heat} \times \Delta T \)

Molar mass is how much one mole of a substance weighs, while specific heat is how much energy is needed per unit of weight. For example, the molar mass of octane \( (C_8H_{18}) \) is approximately 114 g/mol, and its specific heat is 2.22 J/g-K. On the other hand, water has a molar mass of about 18 g/mol and a higher specific heat of 4.18 J/g-K.
When you calculate the energy needed to change the temperature of one mole of each substance by the same amount, you see that:

• Octane: \( 114 \times 2.22 = 253.08 \)
• Water: \( 18 \times 4.18 = 75.24 \)

This means octane requires more heat than water. Comparing molar masses and specific heat helps in predicting how different substances respond to thermal changes, crucial in applications ranging from chemical processes to daily life heating needs.