/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 If a nonhuman civilization were ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 8

If a nonhuman civilization were to develop on Saturn's largest moon, Titan, its scientists might well devise a temperature scale based on the properties of methane, which is much more abundant on the surface than water is. Methane freezes at \(-182.6^{\circ} \mathrm{C}\) on Titan, and boils at \(-155.2^{\circ} \mathrm{C}\) Taking the boiling point of methane as \(100.0^{\circ} \mathrm{M}\) (degrees Methane) and its freezing point as \(0^{\circ} \mathrm{M},\) what temperature on the Methane scale corresponds to the absolute zero point of the Kelvin scale?

Short Answer

Expert verified
Absolute zero is \(-330.51^{\circ} \mathrm{M}\).

Step by step solution

01

Identify the Known Points

We have two known points on the Kelvin scale and the Methane scale: - Methane freezes at \(-182.6^{\circ} \mathrm{C}\), which we'll label as \(T_C = -182.6^{\circ} \mathrm{C}\) and \(T_M = 0^{\circ}\mathrm{M}\).- Methane boils at \(-155.2^{\circ} \mathrm{C}\), or \(T_C = -155.2^{\circ} \mathrm{C}\) and \(T_M = 100.0^{\circ} \mathrm{M}\).
02

Convert Celsius to Kelvin

To convert from Celsius to Kelvin, add 273.15: 1. \(-182.6^{\circ} \mathrm{C} + 273.15 = 90.55\,\mathrm{K}\)2. \(-155.2^{\circ} \mathrm{C} + 273.15 = 117.95\,\mathrm{K}\)
03

Establish a Linear Relationship

The temperature conversion from Kelvin to Methane can be described by a linear equation. Assuming the form \[ T_M = aT_K + b \]where:- \(T_M\) is the temperature in the Methane scale.- \(T_K\) is the temperature in Kelvin.
04

Solve for Constants a and b

Using the two points:1. At freezing: - \(0 = a(90.55) + b\) - This gives: \(b = -90.55a\)2. At boiling: - \(100 = a(117.95) + b\) - Replacing \(b\) from first equation: \[ 100 = a(117.95) - 90.55a \] - Simplify: \[ 100 = 27.4a \] - Solve for \(a\): \[ a = \frac{100}{27.4} = 3.65 \] Substitute \(a = 3.65\) back into \(b = -90.55a\): \[ b = -90.55 \times 3.65 = -330.51 \]
05

Calculate Absolute Zero in Methane Scale

Absolute zero in Kelvin is \(0\,\mathrm{K}\). Plug this into the equation: \[ T_M = 3.65 \times 0 + (-330.51) = -330.51 \]The temperature corresponding to absolute zero on the Methane scale is \(-330.51^{\circ} \mathrm{M}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Methane Scale
Imagine a world where methane, not water, defines the temperature scale. This is what scientists on Titan, Saturn鈥檚 largest moon, might do. Methane is far more plentiful there than water. So, if they set freezing point of methane at \(0^{\circ} \mathrm{M}\) and boiling point at \(100^{\circ} \mathrm{M}\), it provides a familiar structure to Celsius but with methane's unique traits.

On this scale,
  • The freezing point of methane is \( -182.6^{\circ} \mathrm{C} \), corresponding to \( 0^{\circ} \mathrm{M} \).
  • The boiling point of methane is \( -155.2^{\circ} \mathrm{C} \), or \( 100^{\circ} \mathrm{M} \).
This creates a complete new temperature framework based on methane. It allows scientists to describe their environment in a way that naturally aligns with their surroundings.
Kelvin to Methane Conversion
Converting Kelvin to the Methane scale involves understanding the relationship between these scales. We use a simple linear equation model to find the conversion. The line equation is \( T_M = aT_K + b \), representing the conversion.

To determine \(a\) and \(b\):
  • At freezing, \( T_M = 0 \), \( T_K = 90.55 \) K, gives us the first condition.
  • At boiling, \( T_M = 100 \), \( T_K = 117.95 \) K, gives us the second condition.
Solving these provides:
  • \(a = \frac{100}{27.4} = 3.65\)
  • \(b = -90.55 \times 3.65 = -330.51\)
This results in the conversion formula \( T_M = 3.65T_K - 330.51 \), allowing translation between Kelvin and Methane scales seamlessly.
Absolute Zero
Absolute zero represents the theoretical lowest possible temperature. It鈥檚 the point where all atomic motion ceases. On the Kelvin scale, absolute zero is \(0\, \mathrm{K}\).

Now, when we use the conversion formula \( T_M = 3.65 \times 0 - 330.51 \), we find that absolute zero corresponds to \( -330.51^{\circ} \mathrm{M} \) on the Methane scale. This effectively indicates a point of zero thermal energy within this new context.

Understanding absolute zero in different contexts helps illustrate the universality of temperature's fundamental nature, no matter the components defining the scale. It鈥檚 still a key concept linking science across any form of intelligent measurement.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A person eats a container of strawberry yogurt. The Nutritional Facts label states that it contains 240 Calories ( 1 Calorie \(=4186\) J). What mass of perspiration would one have to lose to get rid of this energy? At body temperature, the latent heat of vaporization of water is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\)

One January morning in \(1943,\) a warm chinook wind rapidly raised the temperature in Spearfish, South Dakota, from below freezing to \(+12.0^{\circ} \mathrm{C}\). As the chinook died away, the temperature fell to \(-20.0^{\circ} \mathrm{C}\) in 27.0 minutes. Suppose that a \(19-\mathrm{m}\) aluminum flagpole were subjected to this temperature change. Find the average speed at which its height would decrease, assuming the flagpole responded instantaneously to the changing temperature.

During an all-night cram session, a student heats up a one-half liter \(\left(0.50 \times 10^{-3} \mathrm{m}^{3}\right)\) glass (Pyrex) beaker of cold coffee. Initially, the temperature is \(18^{\circ} \mathrm{C},\) and the beaker is filled to the brim. A short time later when the student returns, the temperature has risen to \(92^{\circ} \mathrm{C} .\) The coefficient of volume expansion of coffee is the same as that of water. How much coffee (in cubic meters) has spilled out of the beaker?

At a fabrication plant, a hot metal forging has a mass of \(75 \mathrm{kg}\) and a specific heat capacity of \(430 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) To harden it, the forging is immersed in \(710 \mathrm{kg}\) of oil that has a temperature of \(32^{\circ} \mathrm{C}\) and a specific heat capacity of \(2700 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) The final temperature of the oil and forging at thermal equilibrium is \(47^{\circ} \mathrm{C}\). Assuming that heat flows only between the forging and the oil, determine the initial temperature of the forging.

One rod is made from lead and another from quartz. The rods are heated and experience the same change in temperature. The change in length of each rod is the same. If the initial length of the lead rod is \(0.10 \mathrm{m},\) what is the initial length of the quartz rod?
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Medical Physics

Read Explanation

Translational Dynamics

Read Explanation

Magnetism

Read Explanation

Dynamics

Read Explanation

Particle Model of Matter

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.