Question

General Chemistry For a certain ideal gas, the volume \(V\) (in cubic centimeters) equals 20 times the temperature \(\underline{T}\) (in degrees Celsius). If the temperature varies from \(80^{\circ}\) to \(120^{\circ} \mathrm{C}\), inclusive, what is the corresponding range of the volume of the gas?

Short answer

The volume ranges from 1600 cm³ to 2400 cm³.

Step-by-step solution

Understand the relationship

The problem states that the volume of the ideal gas is directly proportional to its temperature. The relationship can be written as: \[ V = 20T \] where \( V \) is the volume in cubic centimeters and \( T \) is the temperature in degrees Celsius.

Identify the range of temperature

The temperature \( T \) varies from 80°C to 120°C inclusive. Therefore, \( T \) can be any value between 80 and 120.

Calculate the minimum volume

To find the minimum volume, substitute the lowest temperature value into the relationship: \[ V_{min} = 20 \times 80 = 1600 \] Thus, the minimum volume is 1600 cubic centimeters.

Calculate the maximum volume

To find the maximum volume, substitute the highest temperature value into the relationship: \[ V_{max} = 20 \times 120 = 2400 \] Thus, the maximum volume is 2400 cubic centimeters.

State the range of volume

The volume of the gas varies from the minimum volume to the maximum volume. Hence the range of the volume of the gas is from 1600 cubic centimeters to 2400 cubic centimeters.

Key concepts

volume-temperature relationship

In this exercise, we explore the volume-temperature relationship of an ideal gas. The volume of the gas (\(V\)) is directly related to its temperature (\(T\)). Specifically, the given relationship is: \[ V = 20T \]
Here, volume increases when temperature rises because both are directly proportional.
This direct proportionality means that if temperature doubles, volume also doubles. Conversely, if temperature halves, volume does too.
Understanding this relationship is crucial for predicting gas behavior under different temperatures. When dealing with ideal gases, this relationship simplifies calculations and helps us understand changes in gas properties.

direct proportionality

Direct proportionality is a fundamental concept in science and mathematics. In this exercise, volume (\(V\)) is directly proportional to temperature (\(T\)).
This relationship is expressed as: \[ V = kT \]
In the given problem, the constant (\(k\)) is 20, so we have: \[ V = 20T \]
Direct proportionality implies a linear relationship where one variable changes at a constant rate with another.
This makes it straightforward to predict changes in one quantity based on changes in another.
This consistency is beneficial in various real-world applications, including thermodynamics and gas laws.

range calculation

Range calculation involves determining the difference between the highest and lowest values of a variable. Here, we calculate the range of volume for the ideal gas as temperature varies from 80°C to 120°C.
First, find the minimum volume at 80°C: \[V_{min} = 20 \times 80 = 1600 \text{ cubic centimeters}\right) \]
Next, find the maximum volume at 120°C: \[ V_{max} = 20 \times 120 = 2400 \text{ cubic centimeters} \]
The range of volume is the interval between minimum and maximum volumes. Therefore, the range is from 1600 to 2400 cubic centimeters.
These calculations help us understand the potential extent of volume changes based on temperature variations, offering valuable insights into gas behavior in different environments.