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What mass of \(\mathrm{Al}_2\mathrm{O}_3\) is produced from 18.5 g Al? \(4\mathrm{Al} + 3\mathrm{O}_2 \rightarrow 2\mathrm{Al}_2\mathrm{O}_3\)
Molar mass Al = 27 g/mol: moles Al = 18.5/27 = 0.685 mol. From stoichiometry: 4 mol Al β 2 mol AlβOβ, so mol AlβOβ = 0.685/2 = 0.3425 mol. Molar mass AlβOβ = 102 g/mol: mass = 0.3425Γ102 = 34.94 g β 34.9 g.
On the blink
What is the primary outcome of crossing over during prophase I of meiosis?
Crossing over exchanges genetic material between non-sister chromatids of homologous chromosomes, producing new allele combinations (recombination) and increasing genetic variation.
Let $S=\left\{t\in\mathbb{R}: f(x)=|x-\pi|\cdot(e^{|x|}-1)\sin|x|\right.$ is not differentiable at $t\}$. Find $S$.
Check $x=0$: $f(x)=|x-\pi|(e^{|x|}-1)\sin|x|$. Near $x=0$, both $(e^{|x|}-1)$ and $\sin|x|$ have factors of $|x|$, so $f(x)\approx |x-\pi|\cdot|x|^2$ near 0, which is differentiable at 0.
Check $x=\pi$: near $\pi$, $(e^{|x|}-1)\sin|x|$ is smooth and non-zero ($e^\pi\sin\pi=0$!). Since $\sin\pi=0$, the product $(e^{|x|}-1)\sin|x|$ vanishes at $\pi$, making $f(x)=0$ near $\pi$ β wait, $\sin\pi=0$, so $f(\pi)=0$. Checking differentiability more carefully shows $f$ is differentiable at $\pi$ too.
Therefore $S=\boldsymbol{\emptyset}$.
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