Problem 45 - A Complete Resource Book in Chemistry for JEE Main

Question

The molar heat capacity of water at constant pressure, \(\mathrm{C}\), is \(75 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\). When \(1.0 \mathrm{~kJ}\) of heat is supplied to \(100 \mathrm{~g}\) of water which is free to expand, the increase in temperature of water is (a) \(4.8 \mathrm{~K}\) (b) \(6.6 \mathrm{~K}\) (c) \(1.2 \mathrm{~K}\) (d) \(2.4 \mathrm{~K}\)

Step-by-Step Solution

Answer

(d) 2.4 K

Step 1: Convert mass to moles

We need to convert the given mass of water into moles to use in calculations. The molar mass of water is 18 g/mol. Therefore, the number of moles \( n \) can be calculated as: \[ n = \frac{100 \text{ g}}{18 \text{ g/mol}} \approx 5.56 \text{ moles} \]

Step 2: Calculate heat capacity for given moles

Using the molar heat capacity \( C \) of water, the total heat capacity \( C_t \) for 5.56 moles of water is: \[ C_t = n \times C = 5.56 \text{ moles} \times 75 \text{ J K}^{-1} \text{ mol}^{-1} \approx 417 \text{ J K}^{-1} \]

Step 3: Convert supplied heat to SI units

The heat supplied is given as \(1.0 \text{ kJ}\), which needs to be converted to joules: \[ 1.0 \text{ kJ} = 1000 \text{ J} \]

Step 4: Calculate the temperature increase

Use the formula \( Q = C_t \times \Delta T \) to calculate the increase in temperature \( \Delta T \). Rearranging gives: \[ \Delta T = \frac{Q}{C_t} = \frac{1000 \text{ J}}{417 \text{ J K}^{-1}} \approx 2.4 \text{ K} \]

Step 5: Identify the correct option

The calculated temperature increase \( \Delta T = 2.4 \text{ K} \) matches option (d).

Key Concepts

Molar Heat Capacity

Molar heat capacity is a property that indicates how much heat is needed to raise the temperature of one mole of a substance by one Kelvin. It is essential when dealing with thermodynamic processes as it allows us to calculate how substances react to added or removed heat.

• The unit of molar heat capacity is usually expressed in joules per Kelvin per mole \( (\text{J K}^{-1} \text{ mol}^{-1}) \).
• In the exercise, the molar heat capacity of water is given as \(75 \text{J K}^{-1} \text{ mol}^{-1}\).

By understanding molar heat capacity, we can predict how a given quantity of a substance will respond under heating or cooling. It becomes especially useful when calculating temperature changes for chemical reactions or physical processes in one mole of a material.

Specific Heat Capacity

Specific heat capacity refers to the amount of heat required to raise the temperature of one gram of a substance by one Kelvin. It is a universal concept used to understand how different materials interact with heat. Specific heat capacity is vital in calculating temperature changes in everyday scenarios like heating water or cooking.

• The unit of specific heat capacity is typically joules per gram per Kelvin \( (\text{J g}^{-1} \text{ K}^{-1}) \).
• This property varies between substances. Thus, water has a different specific heat capacity than metals or gases.

While we focused on molar heat capacity in our problem, specific heat is similarly important in thermodynamics by helping determine energy transfers in multi-step reactions or processes in which mass is a variable.

Temperature Change Calculations

Calculating temperature changes involves understanding the basic formula that heat exchanged \( (Q) \) is the product of the heat capacity \( (C_t) \) and the temperature change \( (\Delta T) \). This calculation helps us determine how much a substance's temperature will change when a specific amount of heat is supplied or removed.

• The formula used is \( Q = C_t \times \Delta T \), which can be rearranged to \( \Delta T = \frac{Q}{C_t} \).
• In the exercise, after converting the supplied energy to joules and calculating the total heat capacity, this formula allowed us to find the temperature increase.

By using this method, we can easily figure out how processes like boiling water or cooling in refrigerators occur, and more importantly, predict the outcome of adding or releasing energy to a system.