Question

The temperature of a gas at pressure \(\mathrm{P}\) and volume \(\mathrm{V}\) is \(27^{\circ} \mathrm{C}\) Keeping its volume constant if its temperature is raised to \(927^{\circ} \mathrm{C}\), then its pressure will be (A) \(3 \mathrm{P}\) (B) \(2 \mathrm{P}\) (C) \(4 \mathrm{P}\) (D) \(6 \mathrm{P}\)

Short answer

The new pressure of the gas when its temperature is raised from \(27^{\circ} \mathrm{C}\) to \(927^{\circ} \mathrm{C}\) while keeping the volume constant is \(4 \mathrm{P}\). Therefore, the correct answer is (C) \(4 \mathrm{P}\).

Step-by-step solution

Temperature conversion from Celsius to Kelvin

First, we need to convert both the initial and final temperatures from Celsius to Kelvin, using the relation: \(T_{K} = T_{^{\circ}C} + 273.15\) Convert the initial temperature (\(27^{\circ} \mathrm{C}\)): \(T_1 = 27+ 273.15 = 300.15K\) Convert the final temperature (\(927^{\circ} \mathrm{C}\)): \(T_2 = 927 + 273.15 = 1200.15K\)

Apply the ideal gas law

The ideal gas law equation is: \(PV = nRT\) Since volume and the gas constant (R) do not change, we can compare the pressure and temperature ratios: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) We have already converted the temperatures to Kelvin, and we are looking for the new pressure \(P_2\).

Solve for the new pressure \(P_2\)

Now, we can substitute the known values into the equation: \(\frac{P}{300.15} = \frac{P_2}{1200.15}\) Solve for the new pressure, \(P_2\): \(P_2 = \frac{1200.15}{300.15}P\) \(P_2 = 4P\)

Choose the correct answer

Based on our calculations, the new pressure of the gas is \(4P\). Therefore, the correct answer is (C) \(4 \mathrm{P}\).

Key concepts

Temperature Conversion

Understanding temperature conversion is crucial in solving gas law problems. When dealing with gases, the standard unit for temperature is Kelvin, not Celsius. The reason is that Kelvin is the absolute temperature scale which aligns perfectly with physical metrics used in thermodynamics. For instance, pressure-volume calculations under the ideal gas law are more accurate when temperatures are expressed in Kelvin.
To convert a Celsius temperature to Kelvin, use the conversion formula. Simply add 273.15 to the Celsius temperature. Here's why:

• The Kelvin scale is linked directly to the Celsius scale. Zero on this scale is absolute zero, the coldest possible temperature.
• This conversion is straightforward, making it ideal for scientific calculations.

In our example, we converted two temperatures: 27°C became 300.15K, and 927°C became 1200.15K. This conversion facilitates accurate use of the ideal gas law.

Pressure-Temperature Relationship

The pressure-temperature relationship is a key principle of the ideal gas law. It states that pressure of a given gas is directly proportional to its temperature when volume and moles of gas remain constant. Our exercise maintains a constant volume, making this relationship very useful.

• This principle allows predictions for changes in pressure, given a change in temperature.
• Mathematically, it is expressed through the formula \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), implying that as temperature increases, pressure increases proportionately if the volume is unchanged.

Using this relationship, we can calculate the new pressure of the gas once the temperature rises, by rearranging the formula to solve for the unknown pressure. This understanding directly led us to discover the factor by which pressure increases.

Celsius to Kelvin Conversion

Converting Celsius to Kelvin is an essential step in many chemical and physical computations. It allows temperature to be expressed on an absolute scale that simplifies gas law calculations. Here's how to do it:

• The formula is straightforward: Add 273.15 to degrees Celsius to obtain Kelvin.
• This adjustment accounts for the zero point difference between the Celsius and Kelvin scales.

For this exercise, recognizing how to convert 27°C and 927°C to Kelvin was pivotal. Without this conversion, it would have been complicated to apply the ideal gas law equation accurately.
Knowing the Kelvin temperature is crucial because the effect of temperature changes on pressure is deduced using this unit. This step is foundational in understanding how gases behave under different conditions. Thus, accurate conversion is not just a mathematical need, but a scientific necessity.