Question

At \(\mathrm{O}^{\circ} \mathrm{C}\) the density of a fixed mass of a gas divided by pressure is \(\mathrm{x} .\) At \(100^{\circ} \mathrm{C}\), the ratio will be (A) \(\mathrm{x}\) (B) \([(273) /(373)] \mathrm{x}\) (C) \([(373) /(273)] \mathrm{x}\) (D) \([(100) /(273)] \mathrm{x}\)

Short answer

The short answer is: \( \frac{ρ_2}{P_2} = \frac{373}{273}x \) So, the correct answer is: (C) \([(373) /(273)] \mathrm{x}\)

Step-by-step solution

Recall the ideal gas law

The ideal gas law can be written as: \( PV = nRT \) Where: P = pressure V = volume n = number of moles R = ideal gas constant T = temperature (in Kelvin)

Express density in terms of the ideal gas law

Density (ρ) can be defined as mass (m) divided by volume (V). For a fixed mass of gas, we can express the number of moles (n) as mass (m) divided by the molar mass (M). Thus, n = m/M. Now we can rewrite the ideal gas law in terms of mass (m), molar mass (M), and density (ρ): \( P(V) = \frac{m}{M} RT \) Now, rearrange for the density: \( ρ = \frac{m}{V} = \frac{P M}{RT} \)

Divide both sides of the equation by pressure (P)

We want to get the density divided by pressure (ρ/P). Therefore, divide the equation from step 2 by P. \( \frac{ρ}{P} = \frac{M}{RT} \)

Plug in the values for 0°C and 100°C

Now, let's plug in the values for T = 0°C and T = 100°C and find the ratios. T1 = 0°C = 273 K T2 = 100°C = 373 K At 0°C: \( \frac{ρ_1}{P_1} = \frac{M}{R × 273} \) , where ρ1/P1 corresponds to x. At 100°C: \( \frac{ρ_2}{P_2} = \frac{M}{R × 373} \) Now, let's find the ratio of these expressions. Divide the expression at 100°C by the expression at 0°C: \( \frac{\frac{ρ_2}{P_2}}{\frac{ρ_1}{P_1}} = \frac{\frac{M}{R × 373}}{\frac{M}{R × 273}} \)

Simplify the expression

Cancel out M and R in the expression above: \( \frac{\frac{ρ_2}{P_2}}{x} = \frac{373}{273} \) Now, isolate the term at 100°C: \( \frac{ρ_2}{P_2} = \frac{373}{273}x \) So, the correct answer is: (C) \([(373) /(273)] \mathrm{x}\)

Key concepts

Density and Pressure Relationship

The relationship between density and pressure in gases is closely linked to the ideal gas law. To understand this, let's start with the basics. Density \((\rho)\) is defined as the mass \((m)\) of an object divided by its volume \((V)\). For gases, this means that:
\[ \rho = \frac{m}{V} \]
Now, using the ideal gas law:
\[ PV = nRT \]
where \(P\) stands for pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin. The number of moles \(n\) can also be expressed as the mass \(m\) divided by the molar mass \(M\). Therefore, \(n = \frac{m}{M}\). Substituting this into the ideal gas law formula gives us:
\[ PV = \frac{m}{M}RT \]
Rearranging the equation to solve for density and dividing by pressure \(P\), we develop the equation:
\[ \frac{\rho}{P} = \frac{M}{RT} \]
This equation demonstrates that density divided by pressure is inversely proportional to temperature, if the molar mass is held constant. As a result, as the temperature increases, \(\rho/P\) decreases, assuming pressure remains constant. This derivation is critical for predicting how gases behave under various conditions.

Temperature Conversion to Kelvin

Temperature is crucial in gas behavior analysis, often requiring conversion to the Kelvin scale, as this relates directly to the absolute energy of molecules. The Kelvin scale starts at absolute zero, where theoretically molecular motion stops. To convert Celsius to Kelvin, you simply add 273.15 (often approximated as 273 for simplicity in problem solving).
For example, to convert \(0^{\circ}C\) to Kelvin:
\[ T(\text{K}) = 0 + 273 = 273 \text{ K} \]
And to convert \(100^{\circ}C\):
\[ T(\text{K}) = 100 + 273 = 373 \text{ K} \]
These conversions are essential because gas laws, like the ideal gas law, require absolute temperatures measured in Kelvin to accurately predict how gases behave under different conditions.

Gas Laws in Thermodynamics

The gas laws serve as fundamental principles in thermodynamics, crucial for understanding the behavior of gases. The most prominent among them is the Ideal Gas Law represented as:
\[ PV = nRT \]
This law is a unifying equation, encompassing the behavior described by more specific gas laws such as Boyle’s Law, Charles’s Law, and Avogadro’s Law.

• **Boyle's Law:** At constant temperature, the pressure and volume of a gas are inversely proportional. \(P_1V_1 = P_2V_2\)
• **Charles's Law:** At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\)
• **Avogadro's Law:** For a given temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas. \(V = kn\)

When combined, these laws help predict changes in state or conditions for a gas. By understanding these principles, we gain valuable insights into thermal systems, addressing real-world problems ranging from weather forecasting to engine designs.