Chapter 10: Problem 53
A certain amount of gas at \(25^{\circ} \mathrm{C}\) and at a pressure of 0.800 atm is contained in a vessel. Suppose that the vessel can withstand a pressure no higher than \(5.00 \mathrm{~atm} .\) How high can you raise the temperature of the gas without bursting the vessel?
Short Answer
Expert verified
The temperature can be increased to approximately 1589°C.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gas Pressure
Gas pressure is a fundamental concept in understanding how gases behave under different conditions. It is the force that the gas molecules exert when they collide with the walls of their container. Pressure is measured in various units, such as atmospheres (atm), pascals (Pa), or millimeters of mercury (mmHg). In the context of the Ideal Gas Law, pressure is directly related to temperature and volume.
When the volume of a gas is kept constant, as in a closed container, any increase in temperature leads to an increase in pressure. This is because the gas molecules move more rapidly, colliding with the container walls more frequently and with greater force. This relationship is a direct application of the Ideal Gas Law in its combined form, where pressure divided by temperature remains constant for a given amount of gas.
By understanding how pressure depends on temperature, one can predict how changes in temperature will affect pressure levels, an important aspect when dealing with pressurized vessels.
When the volume of a gas is kept constant, as in a closed container, any increase in temperature leads to an increase in pressure. This is because the gas molecules move more rapidly, colliding with the container walls more frequently and with greater force. This relationship is a direct application of the Ideal Gas Law in its combined form, where pressure divided by temperature remains constant for a given amount of gas.
By understanding how pressure depends on temperature, one can predict how changes in temperature will affect pressure levels, an important aspect when dealing with pressurized vessels.
Temperature Conversion
Temperature conversion is a crucial part of solving problems involving gases, particularly when applying the Ideal Gas Law. Different temperature scales are used depending on the context, and understanding how to switch between them is essential.
For accurate calculations in physics and chemistry, temperatures must be converted to Kelvin. This conversion eliminates negative values and allows for simple proportionate relationships in formulas like the Ideal Gas Law.
- The Celsius scale is commonly used in everyday situations and scientific measurements.
- The Kelvin scale, which starts at absolute zero, is the standard for scientific calculations involving gas laws.
For accurate calculations in physics and chemistry, temperatures must be converted to Kelvin. This conversion eliminates negative values and allows for simple proportionate relationships in formulas like the Ideal Gas Law.
Kelvin Scale
The Kelvin scale is the temperature measurement system used in scientific calculations when working with gases and many other fields. It is an absolute scale, meaning it starts from absolute zero, the theoretical point where all molecular motion ceases. Because of its starting point at absolute zero, the Kelvin scale provides a direct measure of the thermal energy within a system.
One key advantage of using the Kelvin scale is its ability to simplify mathematical equations involving temperature. Since Kelvin temperatures are always positive, they avoid the complications associated with negative temperatures, making them ideal for computations like those in the Ideal Gas Law.
In this exercise, we saw the relevance of Kelvin in calculating how gas pressure responds to temperature changes, ensuring the accuracy of these computations by keeping everything on the universal reference point of this scale.
One key advantage of using the Kelvin scale is its ability to simplify mathematical equations involving temperature. Since Kelvin temperatures are always positive, they avoid the complications associated with negative temperatures, making them ideal for computations like those in the Ideal Gas Law.
In this exercise, we saw the relevance of Kelvin in calculating how gas pressure responds to temperature changes, ensuring the accuracy of these computations by keeping everything on the universal reference point of this scale.
Celsius to Kelvin Conversion
Converting Celsius to Kelvin is a straightforward process but an essential one for many scientific calculations. The conversion formula is simply:\[ T(K) = T(^{\circ}C) + 273.15 \]
This formula is used because the Celsius scale is offset from the Kelvin scale by 273.15 degrees.
For our problem, converting the initial temperature from Celsius (\(25^{\circ}C\)) to Kelvin was a critical step. It gave us \(298.15\, K\), ensuring that our calculations using the Ideal Gas Law were accurate, as these laws assume temperature is expressed on an absolute scale.
This conversion is crucial not just for gas calculations but also in many other scientific contexts, reinforcing the global scientific preference for the Kelvin scale.
This formula is used because the Celsius scale is offset from the Kelvin scale by 273.15 degrees.
For our problem, converting the initial temperature from Celsius (\(25^{\circ}C\)) to Kelvin was a critical step. It gave us \(298.15\, K\), ensuring that our calculations using the Ideal Gas Law were accurate, as these laws assume temperature is expressed on an absolute scale.
This conversion is crucial not just for gas calculations but also in many other scientific contexts, reinforcing the global scientific preference for the Kelvin scale.
