Chapter 18 Multiple choice questions

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. G is a function of temperature T, pressure p, and number of moles n. Write an expression for the partial differential of G with respect to p.

${\left(\frac{\partial G}{\partial p}\right)}_{T,n}$ correct incorrect

$\frac{\partial G}{\partial n}$ correct incorrect

${\left(\frac{\partial G}{\partial p}\right)}_n$ correct incorrect

${\left(\frac{\partial G}{\partial T}\right)}_{p,n}$ correct incorrect

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. Write down the partial differential ${\left(\frac{\partial f}{\partial x}\right)}_{y,z}$of the following function:
f(x, y, z) = 4x⁵ - 3x⁴y + 2x²y²z + 6xy² - 7yz³.

${\left(\frac{\partial f}{\partial x}\right)}_{y,z}=20x^4-3x^4+4x^{2}yz+12y^2$ correct incorrect

${\left(\frac{\partial f}{\partial x}\right)}_{y,z}=20x^4-12x^{3}y+4x{y}^{2}z+6y^2$ correct incorrect

${\left(\frac{\partial f}{\partial x}\right)}_{y,z}=20x^4-12x^{3}y+4x{y}^2+6xy$ correct incorrect

${\left(\frac{\partial f}{\partial x}\right)}_{y,z}=4x^2{y}^2-21y{z}^2$ correct incorrect

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. Write down the partial differential ${\left(\frac{\partial f}{\partial y}\right)}_{x,z}$of the following function:
f(x, y, z) = 4x⁵ - 3x⁴y + 2x²y²z + 6xy² - 7yz³.

${\left(\frac{\partial f}{\partial y}\right)}_{x,z}=-3x^4+4x^{2}yz+12xy-7z^3$ correct incorrect

${\left(\frac{\partial f}{\partial y}\right)}_{x,z}=4x^2{y}^2-21y{z}^2$ correct incorrect

${\left(\frac{\partial f}{\partial y}\right)}_{x,z}=20x^4-12x^{3}y+\hspace{0.17em}4x{y}^{2}z+6y^2$ correct incorrect

${\left(\frac{\partial f}{\partial y}\right)}_{x,z}=17x^4+\hspace{0.17em}4x^{2}yz+12xy$ correct incorrect

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. Write down the partial differential ${\left(\frac{\partial f}{\partial z}\right)}_{x,y}$of the following function:
f(x, y, z) = 4x⁵ - 3x⁴y + 2x²y²z + 6xy² - 7yz³.

${\left(\frac{\partial f}{\partial z}\right)}_{x,y}=20x^4-12x^{3}y+4x{y}^{2}z+6y^2$ correct incorrect

${\left(\frac{\partial f}{\partial z}\right)}_{x,y}=-3x^4+4x^{2}yz+12xy-7z^3$ correct incorrect

${\left(\frac{\partial f}{\partial z}\right)}_{x,y}=4x^5-3x^4+4x^{2}yz+12xy-7z^3$ correct incorrect

${\left(\frac{\partial f}{\partial z}\right)}_{x,z}=2x^2{y}^2-21y{z}^2$ correct incorrect

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. Select the incorrect partial derivative relationship from the list below.

$(\frac{\partial y}{\partial x}{)}_z=-(\frac{\partial y}{\partial z}{)}_x(\frac{\partial z}{\partial x}{)}_y$ correct incorrect

$(\frac{\partial y}{\partial x}{)}_z=\frac{1}{(\partial x/\partial y{)}_z}$ correct incorrect

$(\frac{\partial f}{\partial x}{)}_z=(\frac{\partial f}{\partial y}{)}_x+\left(\frac{\partial f}{\partial x}{)}_y\right(\frac{\partial y}{\partial x}{)}_z$ correct incorrect

$(\frac{\partial f}{\partial x}{)}_z=(\frac{\partial f}{\partial x}{)}_y+\left(\frac{\partial f}{\partial y}{)}_x\right(\frac{\partial y}{\partial x}{)}_z$ correct incorrect

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. Given that $(\frac{\partial y}{\partial z}{)}_x=3z^2-xy$ and $(\frac{\partial z}{\partial x}{)}_y=2y$,
calculate $(\frac{\partial y}{\partial x}{)}_z$.

$(\frac{\partial y}{\partial x}{)}_z=-6y{z}^2+\hspace{0.17em}2x{y}^2$ correct incorrect

$(\frac{\partial y}{\partial x}{)}_z=\frac{3z^2}{2y}-\frac{x}{2}$ correct incorrect

$(\frac{\partial y}{\partial x}{)}_z=3z^2-xy+2y$ correct incorrect

$(\frac{\partial y}{\partial x}{)}_z=3z^2-xy-2y$ correct incorrect

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. The function J is dependent on a, b, and c.
Write down the total differential.

$dJ=(\frac{\partial J}{\partial a}{)}_{b,c}+(\frac{\partial J}{\partial b}{)}_{a,c}+(\frac{\partial J}{\partial a}{)}_{a,b}$ correct incorrect

$dJ=(\frac{\partial J}{\partial a}{)}_{b,c}dbdc+(\frac{\partial J}{\partial b}{)}_{a,c}dadc+(\frac{\partial J}{\partial c}{)}_{a,b}dadb$ correct incorrect

$dJ=(\frac{\partial J}{\partial a}{)}_{b,c}da+(\frac{\partial J}{\partial b}{)}_{a,c}db+(\frac{\partial J}{\partial c}{)}_{a,b}dc$ correct incorrect

$dJ=(\frac{\partial J}{\partial a}{)}_{b,c}db+(\frac{\partial J}{\partial b}{)}_{a,c}dc+(\frac{\partial J}{\partial c}{)}_{a,b}da$ correct incorrect

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. Write down the partial differential $\left(\frac{{\partial }^{2}f}{\partial y\partial z}\right)$ of the following function:
f(x, y, z) = -5x³y² - 6x²yz + 8xy³z² - 2yz⁵.

$\frac{{\partial }^{2}f}{\partial y\partial z}=-10x^{3}y-6x^{2}z+24x{y}^2{z}^2-2z^5$ correct incorrect

$\frac{{\partial }^{2}f}{\partial y\partial z}=-30x^{2}y-12xz+24y^2{z}^2$ correct incorrect

$\frac{{\partial }^{2}f}{\partial y\partial z}=-10x^3+48xy{z}^2$ correct incorrect

$\frac{{\partial }^{2}f}{\partial y\partial z}=-6x^2+48x{y}^{2}z-10z^4$ correct incorrect

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. Write down the partial differential $\left(\frac{{\partial }^{2}f}{\partial x\partial y}\right)$ of the following function:
f(x, y, z) = -5x³y² - 6x²yz + 8xy³z² - 2yz⁵.

$\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial x\partial y}}=-10x^3+48xy{z}^2$ correct incorrect

$\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial x\partial y}}=-30x^{2}y-12xz+24y^2{z}^2$ correct incorrect

$\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial y\partial z}}=-6x^2+48x{y}^{2}z-10z^4$ correct incorrect

$\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial y\partial z}}=-10x^{3}y-6x^{2}z+24x{y}^2{z}^2-2z^5$ correct incorrect

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. Given that f(x, y, z) = 3x² + 5xy²z + 4y³z³.
Calculate${\nabla }^{2}f$.

${\nabla }^{2}f=6x+5y^{2}z+10xyz+12y^2{z}^3+5x{y}^2+12y^3{z}^2$ correct incorrect

${\nabla }^{2}f=10yz+10xy+36y^2{z}^2+5y^2$ correct incorrect

${\nabla }^{2}f=6+10xz+24y{z}^3+24y^{3}z$ correct incorrect

${\nabla }^{2}f=5y^2+10yz+10xy+36y^2{z}^2$ correct incorrect