Problem 1 - Introductory Chemistry Atoms First

Question

An "empty" \(0.500\) -gallon milk carton sits in a room where the temperature is \(25.20^{\circ} \mathrm{C}\). The barometric pressure is exactly 1 atm. (a) What is the Kelvin temperature of the air inside the carton? (b) What is the pressure in millimeters of mercury of the air inside the carton? (c) What is the volume in liters of the air inside the carton? (1 gallon \(=3.78 \mathrm{~L}\) )

Step-by-Step Solution

Answer

(a) The Kelvin temperature of the air inside the carton is \(298.35\ \mathrm{K}\); (b) the pressure in millimeters of mercury is \(760\ \mathrm{mmHg}\); (c) the volume in liters of the air inside the carton is approximately \(1.89\ \mathrm{L}\).

Step 1: (a) Convert temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, we can use the following formula: \[T_{K} = T_{C} + 273.15\] Where \(T_{K}\) is the temperature in Kelvin, and \(T_{C}\) is the temperature in Celsius. Given \(T_{C} = 25.20^{\circ} \mathrm{C}\), we have: \[T_{K} = 25.20 + 273.15\] Now calculate the temperature in Kelvin: \[T_{K} = 298.35\ \mathrm{K}\]

Step 2: (b) Convert pressure from atm to mmHg

To convert pressure from atm to mmHg (millimeters of mercury), we can use the following conversion factor: 1 atm = 760 mmHg Given the pressure is exactly 1 atm, we have: \[P_{mmHg} = 1 \cdot 760\] Now, calculate the pressure in mmHg: \[P_{mmHg} = 760 \ \mathrm{mmHg}\]

Step 3: (c) Convert volume from gallons to liters

To convert the volume from gallons to liters, we can use the following conversion factor: 1 gallon = 3.78 L Given the volume of the carton is 0.500-gallon, we have: \[V_{L} = 0.500 \cdot 3.78\] Now, calculate the volume in liters: \[V_{L} \approx 1.89 \ \mathrm{L}\] So, the answers are (a) the Kelvin temperature of the air inside the carton is 298.35 K; (b) the pressure in millimeters of mercury is 760 mmHg; (c) the volume in liters of the air inside the carton is approximately 1.89 L.

Key Concepts

Temperature Conversion

Temperature conversion is essential in many scientific calculations. In this context, we are converting the temperature from Celsius to Kelvin. The Kelvin scale is an absolute temperature scale used primarily in scientific research. It's the standard unit of temperature in the International System of Units (SI). To convert from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature given. Remember:

• The Kelvin scale never uses the term "degrees." It's written as Kelvin only, for instance, 298.15 K.
• No negative numbers exist in Kelvin, as it starts from absolute zero, the point at which all thermal motion ceases.

Applying the conversion formula, if the room temperature is 25.20°C, the corresponding Kelvin temperature would be 298.35 K. This helps in ensuring uniformity of measurements, especially in thermodynamics.

Pressure Conversion

Pressure is often measured in various units, depending on the context. The most common unit used in chemistry, and specifically here, is 'atmospheres' (atm). To convert pressure to millimeters of mercury (mmHg), the factor used is:

• 1 atm = 760 mmHg

This conversion is crucial when interfacing with older instruments or standards that still use mmHg. Knowing that the pressure of the room is 1 atm, it converts directly to 760 mmHg. Understanding these conversions ensures you can work effectively with diverse pressure measurements in scientific fields.

Volume Conversion

Volume conversion allows you to understand quantities in different measurement systems. Here, the volume of a milk carton is given in gallons, a unit more commonly used in the United States. However, in scientific contexts, liters (L) are the standard unit. Conversion between these units is straightforward:

• 1 gallon = 3.78 liters

For the given problem, a 0.500-gallon carton converts to approximately 1.89 liters. This understanding is essential for calculations or when documenting experiments that require you to report in SI units.

Kelvin Temperature

The Kelvin temperature scale is important because it starts at absolute zero, where no thermal energy remains in a substance. Scientists prefer this scale for its direct relation to energy measurements. Kelvin temperatures are straightforward conversions from Celsius and do not rely on atmospheric conditions, making them especially useful for physical sciences that require precision.
Converting to Kelvin ensures the temperature values used are independent of water's properties, a significant advantage in chemical reactions and physics equations.

Atmospheric Pressure

The term 'atmospheric pressure' refers to the pressure exerted by the weight of air in the atmosphere above a surface. Measured typically with a barometer, pressure at sea level is about 1 atm, which equals 101.3 kPa or 760 mmHg. Atmospheric pressure can influence weather patterns and altitude-related activities. For chemists, understanding pressure in terms of atm can simplify gas laws equations, as many are expressed with these units. This knowledge allows chemists to relate macroscopic pressure changes to molecular level phenomena efficiently.