Problem 139 - Introductory Chemistry Atoms First

Question

Solve each equation for the underlined quantity: (a) \({ }^{\circ} \mathrm{F}=9 / 5^{\circ} \mathrm{C}+32\) (b) \(P V=n R T\) (c) \(E=h c / \underline{\lambda}\)

Step-by-Step Solution

Answer

\( ^\circ C= \frac{5}{9}({ ^\circ F}-32) \), \( P= \frac{nRT}{V} \), and \( \lambda= \frac{hc}{E} \).

Step 1: Identify the parts of the equations

This step is essential to understand each equation and what each symbol represents. (a) The given equation is the temperature conversion formula between degrees Fahrenheit (°F) and degrees Celsius (°C). (b) The given equation is the ideal gas equation, where P represents pressure, V represents volume, n represents the amount of gas in moles, R is the ideal gas constant, and T is the temperature. (c) The given equation represents the energy of a photon. The energy is E, h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon. Now that we understand each equation, we can begin solving them.

Step 2: Solve the equation (a) for °C

To solve the equation for °C, follow these steps: 1. Subtract 32 from both sides of the equation. 2. Multiply both sides of the equation by 5/9. \( ^\circ C= \frac{5}{9}({ ^\circ F}-32) \)

Step 3: Solve the equation (b) for P

To solve the equation for P, follow these steps: 1. Divide both sides of the equation by V. \( P= \frac{nRT}{V} \)

Step 4: Solve the equation (c) for λ

To solve the equation for λ, follow these steps: 1. Multiply both sides of the equation by λ. 2. Divide both sides of the equation by E. \( \lambda= \frac{hc}{E} \) The solutions for each equation are: (a) \( ^\circ C= \frac{5}{9}({ ^\circ F}-32) \) (b) \( P= \frac{nRT}{V} \) (c) \( \lambda= \frac{hc}{E} \)

Key Concepts

Temperature Conversion

Temperature conversion between degrees Fahrenheit (°F) and degrees Celsius (°C) can seem tricky at first.
But once you know the formula, it's quite simple! The formula used is:

• °F = \( \frac{9}{5} \cdot °C + 32 \)

To convert from °F to °C, you'll rearrange the formula to solve for °C using the steps:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \( \frac{5}{9} \).

This gives us: \[°C = \frac{5}{9}(°F - 32)\]This conversion is essential for scientific calculations, especially in regions that use different temperature scales. Understanding this process can make your science or math classes a lot smoother!

Ideal Gas Law

The Ideal Gas Law is one of the cornerstones of physical chemistry. It combines several physical laws into a single equation:

• \( PV = nRT \)

Here:

• \( P \) is pressure
• \( V \) is volume
• \( n \) is the amount of gas in moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

To solve for pressure \( P \), you can rearrange the equation:\[P = \frac{nRT}{V}\]The Ideal Gas Law is very useful for estimating the behavior of gases under different conditions like temperature or pressure changes.
Just keep in mind that real gases can deviate from this behavior when subjected to extreme conditions, like super high pressures or very low temperatures, making accurate measurements necessary.

Photon Energy

Photon energy is a crucial concept in quantum mechanics, where it describes how light and electromagnetic waves behave. The energy of a photon \( E \) is given by the equation:

• \( E = \frac{hc}{\lambda} \)

In this equation:

• \( E \) is the energy of the photon
• \( h \) is Planck's constant \( ≈ 6.626 \times 10^{-34} \text{Js} \)
• \( c \) is the speed of light \( ≈ 3 \times 10^8 \text{m/s} \)
• \( \lambda \) is the wavelength of the photon

Solving for the wavelength \( \lambda \) gives:\[\lambda = \frac{hc}{E}\]This helps in understanding how different wavelengths correspond to different energy levels, which is one of the key ideas in studying the electromagnetic spectrum.
Different forms of radiation (visible light, X-rays, etc.) vary because of their wavelengths and energies.