Problem 15 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

One mole of oxygen is heated from \(0^{\circ} \mathrm{C}\), at constant pressure, till its volume increased by \(10 \%\). The specific heat of oxygen, under these conditions, is \(0.22 \mathrm{cal} / \mathrm{g}-\mathrm{K}\). The amount of heat required is (a) \(32 \times 0.22 \times 27.3 \times 4.2 \mathrm{~J}\) (b) \(16 \times 0.22 \times 27.3 \times 4.2 \mathrm{~J}\) (c) \(\frac{32 \times 0.22 \times 27.3}{4.2} \mathrm{~J}\) (d) \(\frac{16 \times 0.22 \times 27.3}{4.2} \mathrm{~J}\)

Step-by-Step Solution

Answer

The amount of heat required is represented by option (a), which is 32 g × 0.22 cal/g-K × 27.3 K × 4.18 J/cal.

Step 1: Identify the given parameters

The problem states that one mole of oxygen is heated at constant pressure with a volume increase of 10%. The specific heat of oxygen is given as 0.22 cal/g-K. Note that the atomic mass of oxygen is 16 g/mol, so a mole of oxygen weighs 32 g (since an O2 molecule has two atoms).

Step 2: Convert the volume increase to temperature change

Since the gas is at constant pressure and the volume increases by 10%, this corresponds to a temperature increase of 10% above 0 degrees Celsius. We need to convert this percentage to an actual temperature in Kelvin. Since the problem doesn't give a specific volume, but only a percentage increase, we assume the ideal gas law applies. At constant pressure, a 10% increase in volume implies a 10% increase in temperature. As such, if the initial temperature T is increased by 10%, the final temperature is 1.1T.

Step 3: Calculate the change in temperature in Kelvins

To convert the specific heat capacity to SI units, use the conversion 1 cal = 4.18 J. Multiplying 0.22 cal/g-K by 4.18 J/cal gives the specific heat in J/g-K.

Step 5: Calculate the heat required in joules

Applying the formula Q = mcΔT, where Q is the heat required, m is the mass of the gas, c is the specific heat capacity, and ΔT is the change in temperature, we find Q = 32 g × 0.22 cal/g-K × 27.3 K, and then converting the calories to joules gives Q = 32 × 0.22 × 27.3 × 4.18 J.

Step 6: Compare the calculations with the given options

Comparing the calculation with the given options, the correct option is (a) which corresponds to the calculation of 32 × 0.22 × 27.3 × 4.18 J. Hence, no division by 4.2 as indicated in options (c) and (d), and it is for the entire mole, not half mole as in options (b) and (d).

Key Concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of our specific heat calculation problem, it's essential to understand how heat energy can cause changes in temperature and volume of gases at constant pressure.

In thermodynamics, one of the most crucial principles is the conservation of energy, which implies that energy cannot be created or destroyed, only transferred from one form to another. When we heat oxygen, we are transferring energy to it, resulting in a change in temperature and volume as observed in the exercise.

Moreover, the first law of thermodynamics, which is essentially the law of energy conservation, plays a significant role in understanding how the heat added to the system, in this case, oxygen, converts to internal energy causing a temperature rise.

Heat Capacity

Heat capacity, or thermal capacity, is a physical property that represents the amount of heat energy required to raise the temperature of a substance by a certain amount. The specific heat capacity is the heat capacity per unit mass, which in our exercise is given as 0.22 cal/g-K for oxygen.

The higher the specific heat capacity, the more heat is needed to change the temperature of a substance. Calculating the amount of heat required using the specific heat involves the formula: \( Q = mc\Delta T \), where \( Q \) is the heat in joules, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature in Kelvin.

Understanding heat capacity is crucial in physical chemistry as it influences various processes, including phase changes, reaction kinetics, and thermal stability of substances.

Ideal Gas Law

The ideal gas law is a fundamental equation in physical chemistry that describes the behavior of ideal gases. It is expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

In the given problem, we use the ideal gas law implicitly by assuming that a 10% increase in volume at constant pressure indicates a 10% increase in temperature. This assumption is based on the fact that for an ideal gas at constant pressure, the volume is directly proportional to the temperature (Charles's Law).

The ideal gas law provides a way to relate the physical properties of a gas under ideal conditions, which are usually closely met by gases like oxygen at standard temperatures and pressures.

Physical Chemistry Problems

Solving physical chemistry problems requires a strong grasp of concepts such as thermodynamics, the ideal gas law, and heat capacity. Problems often involve calculations that reveal how these concepts interact in real-world scenarios.

For instance, the given exercise involves calculating the amount of heat required to cause a specific temperature change in oxygen. To solve such problems, clear step-by-step approaches are crucial. One must identify the correct formulas, understand the relationships between variables, and ensure units are consistent throughout the calculation.

Students tackling physical chemistry problems should take care to follow a logical problem-solving process, double-check conversions (e.g., from calories to joules), and compare their calculations against the provided options to verify their accuracy.