---
title: "A student measures the number of undecayed radioactive nuclei \\(N\\) in a sample as a function of elapsed time \\(t\\). The decay of the isotope follows the relationship \\(N(t) = N_0 e^{-\\lambda t}\\), where \\(N_0\\) is the initial number of undecayed nuclei at \\(t = 0\\) and \\(\\lambda\\) is the decay constant. To determine the half-life \\(T_{1/2}\\) of the isotope from the experimental data, the student intends to create a linear graph. Which of the following indicates the quantities that should be plotted on the vertical and horizontal axes to yield a straight line, along with the correct expression for \\(T_{1/2}\\) in terms of the slope \\(m\\) of the resulting line of best fit?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/123467/"
date_modified: "2026-09-28T11:59:51+00:00"
---

# A student measures the number of undecayed radioactive nuclei \(N\) in a sample as a function of elapsed time \(t\). The decay of the isotope follows the relationship \(N(t) = N_0 e^{-\lambda t}\), where \(N_0\) is the initial number of undecayed nuclei at \(t = 0\) and \(\lambda\) is the decay constant. To determine the half-life \(T_{1/2}\) of the isotope from the experimental data, the student intends to create a linear graph. Which of the following indicates the quantities that should be plotted on the vertical and horizontal axes to yield a straight line, along with the correct expression for \(T_{1/2}\) in terms of the slope \(m\) of the resulting line of best fit?

A student measures the number of undecayed radioactive nuclei \(N\) in a sample as a function of elapsed time \(t\). The decay of the isotope follows the relationship \(N(t) = N_0 e^{-\lambda t}\), where \(N_0\) is the initial number of undecayed nuclei at \(t = 0\) and \(\lambda\) is the decay constant. To determine the half-life \(T_{1/2}\) of the isotope from the experimental data, the student intends to create a linear graph. Which of the following indicates the quantities that should be plotted on the vertical and horizontal axes to yield a straight line, along with the correct expression for \(T_{1/2}\) in terms of the slope \(m\) of the resulting line of best fit?

- **A.** Plot \(\dfrac{1}{N}\) on the vertical axis and \(t\) on the horizontal axis; \(T_{1/2} = \dfrac{1}{m N_0}\)
- **B.** Plot \(\ln(N)\) on the vertical axis and \(t\) on the horizontal axis; \(T_{1/2} = -m \ln(2)\)
- **C.** Plot \(\ln(N)\) on the vertical axis and \(\dfrac{1}{t}\) on the horizontal axis; \(T_{1/2} = -\dfrac{\ln(2)}{m}\)
- **D.** Plot \(\ln(N)\) on the vertical axis and \(t\) on the horizontal axis; \(T_{1/2} = -\dfrac{\ln(2)}{m}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123467/*