91影视

The term density may be defined as the ratio of mass and volume.

Consider the given data as below.

The volume of the balloon$V_b=750{\mathrm{m}}^3$

The pressure in the balloon,$P_b=1.01脳{10}^5\mathrm{Pa}$

The volume of filled gas cylinder$V_c=1.90{\mathrm{m}}^3$

TThe molar mass,$M=2.02\mathrm{g}/\mathrm{mol}=2.02脳{10}^{鈭�3}\mathrm{kg}/\mathrm{mol}$

Temperature, $\mathrm{T}={15}^{鈭�}\mathrm{C}=(15+273.15)\mathrm{K}=288.15\mathrm{K}$

The universal gas constant, R = 8.31 J/mol.K

Gravity, $\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$

Air density,$\mathrm{蚁}=1.23\mathrm{kg}/{\mathrm{m}}^3$

The pressure of filled gas cylinder,${\mathrm{P}}_{\mathrm{b}}=1.20脳{10}^6\mathrm{Pa}$

Now let be the number of cylinders. Therefore,

$\begin{array}{r}{\mathrm{P}}_{\mathrm{b}}{\mathrm{V}}_{\mathrm{b}}={\mathrm{nP}}_{\mathrm{c}}{\mathrm{V}}_{\mathrm{c}}\\ \mathrm{n}=\frac{{\mathrm{P}}_{\mathrm{b}}{\mathrm{V}}_{\mathrm{b}}}{{\mathrm{P}}_{\mathrm{c}}{\mathrm{V}}_{\mathrm{c}}}\\ \mathrm{Substitute}\mathrm{known}\mathrm{values}\mathrm{in}\mathrm{the}\mathrm{above}\mathrm{equation},\mathrm{and}\mathrm{you}\mathrm{have}\\ \mathrm{n}=\frac{\left(1.01脳{10}^5\right)\left(750\right)}{\left(1.20脳{10}^6\right)(1.90)}\\ =33.224\\ 鈮�34\end{array}$

Hence, the number of cylinders required is 34.

Now, the number of moles can be written as,

$\mathrm{n}=\frac{\mathrm{m}}{\mathrm{M}}$

Where is the mass of gas given and is the atomic weight.

So, by using ideal gas,

PV = nRT

$\mathrm{PV}=\frac{\mathrm{m}}{\mathrm{M}}\mathrm{RT}$

Therefore,

${\mathrm{m}}_1=\frac{{\mathrm{P}}_{\mathrm{b}}{\mathrm{V}}_{\mathrm{b}}{\mathrm{M}}_1}{\mathrm{RT}}$

Substitute known values in the above formula.

$\begin{array}{r}{\mathrm{m}}_1=\frac{\left(1.01脳{10}^5\right)\left(750\right)\left(2.02脳{10}^{鈭�3}\right)}{\left(8.31\right)\left(288.15\right)}\\ =63.90\mathrm{kg}\end{array}$

Draw the free body diagram as below.

From the above figure,

$\begin{array}{r}{F}_B=m鈰�g+W\\ 蚁Vg=m鈰�g+W\\ W=蚁V_{b}g鈭�m_{1}g\end{array}$

Substitute known values in the above equation.

$\begin{array}{r}W=\left[\right(1.23)脳(750)脳(9.8)鈭�(63.90)脳(9.8\left)\right]\\ =8414.3\mathrm{N}\end{array}$

If the balloon filled with helium gas than using ideal gas equation the mass can be calculated as below.

localid="1668311068246" $$\begin{gathered} {\mathrm{m}}_2=\frac{{\mathrm{P}}_{\mathrm{b}}{\mathrm{V}}_{\mathrm{b}}{\mathrm{M}}_2}{\mathrm{RT}} \\ =\frac{\left(1.01脳{10}^5\right)\left(750\right)\left(4.00脳{10}^{鈭�3}\right)}{(8.31)(288.15)} \\ =126.54\mathrm{kg} \\ \mathrm{Now} \\ {\mathrm{F}}_{\mathrm{B}}=\mathrm{m}鈰�\mathrm{g}+{\mathrm{W}}_{\mathrm{He}} \\ \mathrm{蚁Vg}=\mathrm{m}鈰�\mathrm{g}+{\mathrm{W}}_{\mathrm{He}} \\ \mathrm{W}={\mathrm{蚁V}}_{\mathrm{b}}\mathrm{g}鈭�{\mathrm{m}}_2\mathrm{g} \\ \mathrm{W}=\left[\right(1.23)脳(750)脳(9.8)鈭�(126.54)脳(9.8\left)\right] \\ =7800.4\mathrm{N} \end{gathered}$$

Hence, the weight of helium is 7800.4 N.