Problem 47 - Introductory Chemistry Atoms First

Question

According to the ideal gas law, what would happen to the pressure of a gas if you doubled the amount of gas in a container while also tripling the Kelvin temperature of the gas? Explain.

Step-by-Step Solution

Answer

The pressure of the gas will increase by a factor of 6 if the amount of gas is doubled and the Kelvin temperature is tripled, assuming a constant volume. This is based on the ideal gas law equation \(PV = nRT\), and by applying the given changes, we find that \(P_{new} = 6P\).

Step 1: The ideal gas law is given by the equation: \(PV = nRT\), where P is pressure, V is volume, n is the amount of gas, R is the ideal gas constant, and T is the temperature in Kelvin. #Step 2: Establish the given changes in the gas and temperature#

According to the exercise, we have two changes: 1. The amount of gas (n) is doubled, so the new amount of gas is \(2n\). 2. The Kelvin temperature (T) is tripled, so the new temperature is \(3T\). #Step 3: Apply the changes to the ideal gas law#

Step 2: To find the new pressure after applying the given changes, we can substitute the new values of n and T into the ideal gas law: \(P_{new}V = (2n)R(3T)\). #Step 4: Introduce a constant volume#

Since we're not given any information about how volume (V) is changing, we can assume it remains constant. This means the original and new pressure can be compared at the same volume. We can therefore divide the new equation by the original equation: \(\frac{P_{new}V}{PV} = \frac{(2n)(3T)(R)}{nTR}\). #Step 5: Simplify and solve for the new pressure#

Step 3: Simplify the equation by canceling out common terms: \(\frac{P_{new}}{P} = \frac{6nT}{nT}\). Now, the n and T terms cancel out: \(\frac{P_{new}}{P} = 6\) Finally, solve for the new pressure: \(P_{new} = 6P\) #Step 6: Interpret the result#

The result shows that the pressure of the gas (P) will increase by a factor of 6 if the amount of gas (n) is doubled and the Kelvin temperature (T) is tripled while keeping volume constant.

Key Concepts

Pressure

Pressure is a crucial concept when dealing with gases and their behaviors. In any container, gas molecules move around and collide with the walls of that container. This pushing force per unit area exerted by the gas molecules is what we call pressure. The more frequent and intense these collisions, the higher the pressure.

In physics and chemistry, pressure is generally measured in units such as Pascals (Pa), atmospheres (atm), or millimeters of mercury (mmHg). For any gas, pressure can change depending on several factors, including the amount of gas, volume it occupies, and its temperature.

Understanding pressure is fundamental when applying the ideal gas law, which relates pressure to other gas variables like volume, temperature, and amount of gas. This understanding helps predict how changes in these variables affect gas pressure.

Kelvin temperature

Kelvin is the standard unit of temperature used in the ideal gas law, mainly because it provides an absolute scale of measurement. This means that zero Kelvin (0 K), known as absolute zero, is the point at which all molecular motion theoretically ceases.

The Kelvin scale is essential because gas behaviors are related to the absolute temperature of the gas.

• When the Kelvin temperature of a gas increases, its molecules move faster due to the increase in kinetic energy.
• An increase in temperature typically leads to an increase in pressure, assuming the volume remains constant.

In the ideal gas law, denoted as \(PV = nRT\), temperature (T) must always be in Kelvin. This affects calculations by ensuring that the direct relationships between pressure, volume, and temperature are accurately represented.

Amount of gas

The amount of gas in a system is usually expressed in terms of moles (n), a unit that offers a standardized measure of quantity based on atoms and molecules. Moles, derived from Avogadro's number (\(6.022 \times 10^{23}\) particles per mole), allow for consistent calculations in chemical reactions, including those involving gases.

In the context of the ideal gas law, increasing the amount of gas (n) within a fixed volume often leads to a higher pressure. This occurs because more gas molecules result in more collisions with container walls, thus increasing the pushing force, or pressure.

• Doubling the amount of gas doubles potential collisions.
• In a fixed volume with constant temperature, pressure increases proportionally.

These relationships are crucial in predicting how changes in the amount of gas affect pressure and other variables in the ideal gas law equation.

Ideal gas constant

The ideal gas constant, often denoted as \(R\), plays a pivotal role in the ideal gas law equation \(PV = nRT\). This constant serves as a bridge to relate the properties of temperature, pressure, volume, and the amount of gas in a system.

The value of \(R\) depends on the units used for pressure, volume, and temperature. Commonly, \(R\) is given as:

• 8.314 J/(mol·K) when using metric units (joules, moles, Kelvin)
• 0.0821 L·atm/(mol·K) when using more traditional units (liters, atmospheres, moles, Kelvin)

Understanding \(R\) is essential for accurately calculating and predicting the behavior of ideal gases. Its consistency helps ensure that, regardless of the conditions being measured, pressure, volume, and temperature correlate as expected when using the ideal gas law.