A \(0.50 \mathrm{~L}\) container is occupied by nitrogen at a pressure of 800 torr and a temperature of \(0^{\circ} \mathrm{C}\). The container can only withstand a pressure of \(3.0 \mathrm{~atm} .\) What is the highest temperature \(\left({ }^{\circ} \mathrm{C}\right)\) to which the container may be heated? (a) 505 (b) 450 (c) 625 (d) 560

The ideal gas law is a fundamental principle in chemistry that describes the behavior of ideal gases. It is expressed as the equation PV = nRT, where P is the pressure of the gas, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. Clarity comes from understanding that this relationship provides a way to calculate one variable if the others are known.

In the context of combined gas law problems, the ideal gas law simplifies when we consider a constant quantity of gas in a container that does not change its volume, such as in the textbook problem. Since n (number of moles) and R are constants, and V remains unchanged, the relationship simplifies to \( P/T = k \) (with k being a constant), indicating that the pressure and temperature are directly proportional when volume and the amount of gas are constant.

Gas pressure conversion is essential in solving problems involving different units of measurement for pressure. Common units include atmospheres (atm), torrs, Pascals (Pa), and millimeters of mercury (mmHg).

Key Conversion Factors:

• 1 atm = 760 torr
• 1 atm = 101,325 Pa
• 1 atm = 760 mmHg

To avoid errors, convert all pressure measurements to the same unit before using them in calculations. In the original problem, the pressure given in torrs was successfully converted to atmospheres using the factor \(1 atm = 760 torr\). Without a correct conversion, the final result would have been skewed, leading to the wrong answer. Recognizing the correct unit conversions is a vital skill in chemistry and helps to ensure accurate results in calculations.

Temperature scale conversion is necessary when solving gas law problems because the ideal gas law requires temperatures to be in Kelvin. Kelvin is the SI base unit for temperature. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. Likewise, convert Kelvin to Celsius by subtracting 273.15 from the Kelvin temperature.

This step is critical because failing to use Kelvin could result in incorrect proportions when applying the gas laws. In the given exercise, the initial temperature was given in Celsius and converted to Kelvin, which then allowed the problem to be solved accurately. Always remember to convert back to the required temperature scale, if needed, after performing all gas law calculations in Kelvin.

JEE (Joint Entrance Examination) Physical Chemistry is a crucial subject for engineering aspirants in India, involving a deep understanding of chemical laws and numerical problem-solving abilities. Topics like the ideal gas law, gas pressure, and temperature conversions are standard in the JEE Physical Chemistry syllabus.

Candidates are expected to apply concepts such as the combined gas law, which is a combination of Boyle's law, Charles's law, and Gay-Lussac's law, to solve complex problems. Mastery in converting units, handling standard temperature and pressure conditions, and manipulating equations is fundamental. The original exercise could be representative of a question from the JEE exams, where precision in mathematical operations and a strong grasp of theoretical concepts are tested. As such, both theoretical knowledge and calculation skills are key to succeeding in Physical Chemistry sections of competitive exams like the JEE.